1 Introduction

1.1 The periodic nonlocal Cahn-Hilliard equation

Suppose \(\Omega = (0,L_1)\times (0,L_2) \subset \mathbb {R}^2\). Define \(C^m_\mathrm{per}(\Omega ) = \left\{ f \in C^m\left( \mathbb {R}^2\right) | f \hbox {is } \Omega \hbox {---periodic }\right\} \), where \(m\) is a positive integer. Set \(C^\infty _\mathrm{per}(\Omega ) = \bigcap _{m=1}^\infty C^m_\mathrm{per}(\Omega )\). For any \(1\le q<\infty \), define \(L^q_\mathrm{per}(\Omega ) = \left\{ f \in L_\mathrm{loc}^q\left( \mathbb {R}^2\right) \ | \ f \hbox { is} \Omega \hbox {---periodic}\right\} \). Of course, \(L^q(\Omega )\) and \(L_\mathrm{per}^q(\Omega )\) can be identified in a natural way. Suppose that \(\mathsf {J}:\mathbb {R}^2 \rightarrow \mathbb {R}\) satisfies

A1.:

\(\mathsf {J}= \mathsf {J}_c-\mathsf {J}_e\), where \(\mathsf {J}_c,\, \mathsf {J}_e \in C_\mathrm{per}^\infty \left( \Omega \right) \) are non-negative.

A2.:

\(\mathsf {J}_c\) and \(\mathsf {J}_e\) are even, i.e., \(\mathsf {J}_\alpha (x_1,x_2)\!=\!\mathsf {J}_\alpha (-x_1,-x_2)\), for all \(x_1,x_2\!\in \! \mathbb {R}\), \(\alpha = c\), \(e\).

A3.:

\(\int _\Omega \mathsf {J}(\mathbf{x})\, d\mathbf{x} > 0\).

\(\mathsf {J}\) is called the convolution kernel herein. Given \(\phi \in L_\mathrm{per}^2(\Omega )\), by \(\mathsf {J}*\phi \) we mean the circular, or periodic, convolution defined as

$$\begin{aligned} \left( \mathsf {J}*\phi \right) (\mathbf{x}) := \int \limits _\Omega \mathsf {J}(\mathbf{y}) \, \phi (\mathbf{x}-\mathbf{y}) \, d\mathbf{y} = \int \limits _\Omega \mathsf {J}(\mathbf{x}-\mathbf{y}) \, \phi (\mathbf{y}) \, d\mathbf{y}. \end{aligned}$$
(1.1)

Clearly \(\mathsf {J}*1=\int _\Omega \mathsf {J}(\mathbf{x})\, d\mathbf{x}\) is a positive constant. For any \(\phi \in L_\mathrm{per}^4(\Omega )\), we define

$$\begin{aligned} E(\phi ) = \frac{1}{4}\left\| \phi \right\| _{L^4}^4 + \frac{\gamma _c-\gamma _e+\mathsf {J}*1}{2}\left\| \phi \right\| _{L^2}^2 -\frac{1}{2}\left( \phi , \mathsf {J}*\phi \right) _{L^2} , \end{aligned}$$
(1.2)

where \(\gamma _c,\, \gamma _e \ge 0\) are constants. Typically, one of \(\gamma _c\) and \(\gamma _e\) are zero. The energy (1.2) is equivalent to

$$\begin{aligned} E(\phi ) = \int \limits _\Omega \left\{ \frac{1}{4}\phi ^4+ \frac{\gamma _c-\gamma _e}{2}\phi ^2 + \frac{1}{4}\int \limits _\Omega \mathsf {J}(\mathbf{x}-\mathbf{y})\left( \phi (\mathbf{x})-\phi (\mathbf{y})\right) ^2d\mathbf{y} \right\} d\mathbf{x}.\quad \quad \end{aligned}$$
(1.3)

We have the following properties of the energy, which we state without proof:

Lemma 1.1

There exists a non-negative constant \(C\) such that \(E(\phi ) +C \ge 0\) for all \(\phi \in L_\mathrm{per}^4(\Omega )\). More specifically,

$$\begin{aligned} \frac{1}{8}\left\| \phi \right\| _{L^4}^4&\le E(\phi ) + \frac{\left( \gamma _c-\gamma _e-2\left( \mathsf {J}_e*1\right) \right) ^2}{2}|\Omega |,\end{aligned}$$
(1.4)
$$\begin{aligned} \frac{1}{2}\left\| \phi \right\| _{L^2}^2&\le E(\phi ) +\frac{\left( \gamma -\gamma _e-2\left( \mathsf {J}_e*1\right) -1\right) ^2}{4}|\Omega |. \end{aligned}$$
(1.5)

If \(\gamma _e = 0\), then \(E(\phi ) \ge 0\) for all \(\phi \in L_\mathrm{per}^4(\Omega )\). Furthermore, the energy (1.2) can be written as the difference of convex functionals, i.e, \(E = E_c-E_e\), where

$$\begin{aligned} E_c(\phi )&= \frac{1}{4}\left\| \phi \right\| _{L^4}^4 + \frac{\gamma _c+2\left( \mathsf {J}_c*1\right) }{2}\left\| \phi \right\| _{L^2}^2 ,\end{aligned}$$
(1.6)
$$\begin{aligned} E_e(\phi )&= \frac{\gamma _e+\mathsf {J}_c*1+\mathsf {J}_e*1}{2}\left\| \phi \right\| _{L^2}^2 + \frac{1}{2}\left( \phi , \mathsf {J}*\phi \right) _{L^2}. \end{aligned}$$
(1.7)

The decomposition above is not unique; for instance, one could argue that a more natural convex splitting of the energy is given by

$$\begin{aligned} \tilde{E}_c(\phi )&= \frac{1}{4}\left\| \phi \right\| _{L^4}^4 + \frac{\gamma _c}{2}\left\| \phi \right\| _{L^2}^2 +\frac{\mathsf {J}_c*1}{2}\left\| \phi \right\| _{L^2}^2 -\frac{1}{2}\left( \phi , \mathsf {J}_c*\phi \right) _{L^2}, \\ \tilde{E}_e(\phi )&= \frac{\gamma _e}{2}\left\| \phi \right\| _{L^2}^2 +\frac{\mathsf {J}_e*1}{2}\left\| \phi \right\| _{L^2}^2 -\frac{1}{2}\left( \phi , \mathsf {J}_e*\phi \right) _{L^2}.\\ \end{aligned}$$

But (1.6)–(1.7) has some advantages over this last decomposition. Specifically, (1.6)–(1.7) will allow us to separate the nonlocal and nonlinear terms, treating the nonlinearity implicitly and the nonlocal term explicitly, without sacrificing numerical stability. The chemical potential \(w\) relative to the energy (1.3) is defined as

$$\begin{aligned} w := \delta _{\phi } E = \phi ^3 +\gamma _c \phi -\gamma _e \phi + (\mathsf {J}*1)\phi - \mathsf {J}*\phi . \end{aligned}$$
(1.8)

The conserved gradient dynamics,

$$\begin{aligned} \partial _t \phi = \Delta w , \end{aligned}$$
(1.9)

yields what we will refer to as the nonlocal Cahn-Hilliard (nCH) equation. Formally, periodic solutions of (1.9) dissipate \(E\) at the rate \(d_t E(\phi ) = -\left\| \nabla w \right\| _{L^2}^2\). The nCH equation (1.9) can be rewritten as

$$\begin{aligned} \partial _t \phi = \nabla \cdot \left( a(\phi )\nabla \phi \right) -\left( \Delta \mathsf {J}\right) *\phi \ , \end{aligned}$$
(1.10)

where

$$\begin{aligned} a(\phi ) = 3\phi ^2 +\gamma _c-\gamma _e + \mathsf {J}*1. \end{aligned}$$
(1.11)

We refer to \(a(\phi )\) as the diffusive mobility, or just the diffusivity. To make (1.9) positive diffusive (and non-degenerate) we require [9]

$$\begin{aligned} \gamma _c-\gamma _e +\mathsf {J}*1 =: \gamma _0 > 0 , \end{aligned}$$
(1.12)

in which case \(a(\phi ) >0\). We will assume that (1.12) holds in the sequel. We will not explicitly consider the non-conserved dynamics case herein,

$$\begin{aligned} \partial _t \phi = - w , \end{aligned}$$
(1.13)

though many of the ideas we present will be applicable for this model as well.

1.2 A semi-discrete convex-splitting scheme

Our principal goal in this paper is to describe a first-order accurate (in time) convex splitting scheme that is unconditionally solvable, unconditionally energy stable, and convergent. To motivate the fully discrete scheme that will follow later, we here present semi-discrete version and briefly describe its properties.

A first-order (in time) convex splitting scheme for the nCH Eq. (1.9) can be constructed as follows: given \(\phi ^k\in C^\infty _\mathrm{per}(\Omega )\), find \(\phi ^{k+1},\, w^{k+1}\in C^\infty _\mathrm{per}(\Omega )\) such that

$$\begin{aligned} \phi ^{k+1}-\phi ^k&= s\Delta w^{k+1},\end{aligned}$$
(1.14)
$$\begin{aligned} w^{k+1}&= \left( \phi ^{k+1}\right) ^3 \!+\! \left( 2(\mathsf {J}_c*1)\!+\!\gamma _c\right) \phi ^{k+1} \!-\!\left( \mathsf {J}_c*1+ \mathsf {J}_e*1 +\gamma _e\right) \phi ^k - \mathsf {J}*\phi ^k ,\nonumber \\ \end{aligned}$$
(1.15)

where \(s>0\) is the time step size. Notice that this scheme respects the convex splitting of the energy \(E\) given in (1.6) and (1.7). The contribution to the chemical potential corresponding to the convex energy, \(E_c\), is treated implicitly, the part corresponding to the concave part, \(E_e\), is treated explicitly. Eyre [19] is often credited with proposing the convex splitting methodology for the Cahn-Hilliard and Allen-Cahn equations. The idea is, however, quite general and can be applied to any gradient flow of an energy that splits into convex and concave parts. See for example [37–40]. The method can even be extended to second-order accuracy in time [5, 28, 34].

We have the following a priori energy law for the solutions of the first-order scheme (1.14)–(1.15), which is stated without proof. The proof for the fully discrete version appears later in Section 3.2, and the details are similar.

Theorem 1.2

Suppose the energy \(E(\phi )\) is as defined in Eq. (1.2). For any \(s>0\), the convex splitting scheme (1.14)–(1.15) has a unique solution \(\phi ^{k+1},\, w^{k+1}\in C^\infty _\mathrm{per}(\Omega )\). Moreover, for any \(k\ge 1,\left( \phi ^{k+1}-\phi ^k,1\right) = 0\) and

$$\begin{aligned} E\left( \phi ^{k+1}\right) + s\left\| \nabla w^{k+1} \right\| _{L^2}^2 + R\left( \phi ^k,\phi ^{k+1}\right) = E\left( \phi ^k\right) \ , \end{aligned}$$
(1.16)

where

$$\begin{aligned} R\left( \phi ^k,\phi ^{k+1}\right)&= \frac{1}{4}\left\| \left( \phi ^{k+1}\right) ^2-\left( \phi ^k\right) ^2 \right\| _{L^2}^2 + \frac{1}{2}\left\| \phi ^{k+1}\left( \phi ^{k+1}-\phi ^k\right) \right\| _{L^2}^2 \nonumber \\&+\frac{\left( 3(\mathsf {J}_c*1) + \mathsf {J}_e*1+\gamma _c+\gamma _e\right) }{2}\left\| \phi ^{k+1}-\phi ^k \right\| _{L^2}^2 \nonumber \\&+ \frac{1}{2}\left( \mathsf {J}*\left( \phi ^{k+1}-\phi ^k\right) , \phi ^{k+1}-\phi ^k \right) _{L^2} \end{aligned}$$
(1.17)

The remainder term, \(R\left( \phi ^k,\phi ^{k+1}\right) \), is non-negative, which implies that the energy is non-increasing for any \(s\), i.e., \(E\left( \phi ^{k+1}\right) \le E\left( \phi ^k\right) \), and we say that the scheme is unconditionally energy stable.

In the sequel, we will propose fully discrete versions of this scheme and provide the corresponding analysis.

1.3 Related models and existing numerical results

Equation (1.9) is in the more general family of models:

$$\begin{aligned} \partial _t \phi = \nabla \cdot \left( M(\phi )\nabla w\right) ; \quad w = f'(\phi )+ (\mathsf {J}*1)\phi - \mathsf {J}*\phi . \end{aligned}$$
(1.18)

The corresponding energy for this equation is

$$\begin{aligned} E(\phi ) = \int \limits _\Omega \left( f(\phi ) + \frac{1}{4}\int \limits _\Omega \mathsf {J}(\mathbf{x}-\mathbf{y})\left( \phi (\mathbf{x})-\phi (\mathbf{y})\right) ^2d\mathbf{y} \right) d\mathbf{x}. \end{aligned}$$
(1.19)

With the appropriate choices of mobility \(M\), free energy density \(f\), and interaction kernel \(\mathsf {J}\), Eq. (1.18) resembles the dynamical density functional theory (DDFT) equations. See, e.g., [3, 4, 15, 18, 29, 30, 36]. There are also biological applications for equations of this form [13, 41]. We point out that integro-partial differential equations (IPDE) like Eq. (1.18) also appear in [21, 22, 26, 27, 31, 32] for applications in phase separation and image processing, to name a few. There are a number of analysis results concerning IPDEs like (1.18). Specifically, see [6, 8–10, 20, 22–24, 26] and references therein.

The energy (1.19) can be related to the more familiar (local) Ginzburg-Landau energy

$$\begin{aligned} E_{CH}(\phi ) = \int \limits _\Omega \left( f(\phi ) + \frac{\epsilon ^2}{2}|\nabla \phi |^2\right) d\mathbf{x} , \end{aligned}$$
(1.20)

relative to which the classical Cahn-Hilliard equation [11, 12] is the conserved gradient flow:

$$\begin{aligned} \partial _t\phi = \Delta \left( f'(\phi ) - \epsilon ^2 \Delta \phi \right) . \end{aligned}$$
(1.21)

To see the relationship between the local and nonlocal energies, one approximates the gradient energy density \(\frac{1}{4}\int _\Omega \mathsf {J}(\mathbf{x}-\mathbf{y})\left( \phi (\mathbf{x})-\phi (\mathbf{y})\right) ^2d\mathbf{y}\) using a Taylor expansion [3, 10, 18, 41]. Specifically, assuming \((\phi (\mathbf{x})-\phi (\mathbf{y}))\approx (\mathbf{x}-\mathbf{y})\cdot \nabla \phi (\mathbf{x})\), we find

$$\begin{aligned} \frac{1}{4}\int \mathsf {J}(\mathbf{x}\!-\!\mathbf{y})\left( \phi (\mathbf{x})\!-\!\phi (\mathbf{y})\right) ^2d\mathbf{y}&\approx \frac{1}{4}\int \mathsf {J}(\mathbf{x}\!-\!\mathbf{y})\left( (\mathbf{x}\!-\!\mathbf{y})\cdot \nabla \phi (\mathbf{x}))\right) ^2d\mathbf{y} \!=\! \frac{\epsilon ^2}{2}|\nabla \phi |^2,\nonumber \\ \end{aligned}$$
(1.22)

where \(\epsilon ^2 = \frac{1}{2}\int _\Omega \mathsf {J}(\mathbf{x})|\mathbf{x}|^2 \, d\mathbf{x}\) is the second moment of \(\mathsf {J}\), which is clearly non-negative if \(\mathsf {J}_e \equiv 0\). Periodic solutions of the Cahn-Hilliard equation dissipate the Ginzburg-Landau energy (1.20) at the rate \(d_t E_{CH}(\phi ) = -\left\| \nabla \left( \delta _\phi E_{CH}\right) \right\| _{L^2}^2\), in analogy with the non-local case. The phase field crystal (PFC) equation [16, 17, 39] can be related to DDFT model in a similar manner. In essence, for the PFC equation, given that the kernel \(\mathsf {J}\) in the DDFT model has the appropriate structure, the following fourth order expansion will be a good approximation of the gradient energy [16]:

$$\begin{aligned} \frac{1}{4}\int \limits _\Omega \mathsf {J}(\mathbf{x}-\mathbf{y})\left( \phi (\mathbf{x})-\phi (\mathbf{y})\right) ^2d\mathbf{y} \approx - \frac{\epsilon _2^2}{2}|\nabla \phi |^2 + \frac{\epsilon _4^2}{2}\left( \Delta \phi \right) ^2. \end{aligned}$$
(1.23)

In this case it is usual that \(\mathsf {J}_c\) and \(\mathsf {J}_e\) are both non-trivial, i.e., \(\mathsf {J}_c,\, \mathsf {J}_e\not \equiv 0\).

There are a few works dedicated to numerical simulation of, or numerical methods for, equations like (1.9) and (1.13). Anitescu et al. [2] considered an implicit-explicit time stepping framework for a nonlocal system modeling turbulence, where, as here, the nonlocal term is treated explicitly. The work in [14, 42] address the finite element approximation (in space) of nonlocal peridynamic equations with various boundary conditions. In [7], a finite difference method for Eq. (1.13) with non-periodic boundary conditions is applied and analyzed. The work in [26] uses a spectral-Galerkin method to solve a nonlocal Allen-Cahn type problem, like Eq. (1.13), but with a stochastic noise term and equation modeling heat flow. For other papers focused on approximating solutions to equations like (1.9), see [1, 21, 27, 33].

Our method, which is based on the convex splitting introduced earlier, is the first that we know of that is shown to be unconditionally energy stable, unconditionally uniquely solvable, and convergent for the nCH Eq. (1.9). We use a finite difference method to discretize space, but a galerkin spatial discretization is also possible. We are also able, based on the structure of our implicit-explicit method—specifically, since we are able to separate the nonlinear and nonlocal terms of the scheme—to implement a very efficient nonlinear multigrid solver, though the full details of the performance and implementation will appear only in a later work. The present work focuses on the 2D case but is scalable to 3D easily. Using the a-priori stabilities that we are able to obtain, we prove the convergence for our scheme, in both \(\ell ^2\) and \(\ell ^\infty \) spatial norms. The proofs of our estimates are notable for the fact that they do not require point-wise boundedness of the numerical solution, nor a global Lipschitz assumption or cut-off for the nonlinear term. The \(\ell ^2\) convergence proof requires no refinement path constraint, while the one involving the \(\ell ^\infty \) norm requires only a mild linear refinement constraint, \(s \le C h\).

The key estimates for the error analyses, which are summarized in Sect. 2, take full advantage of the unconditional \(\ell ^\infty (0, T; \ell ^4)\) stability of the numerical solution and an interpolation estimate of the form \(\left\| \phi \right\| _4 \le C \left\| \phi \right\| _2^\alpha \left\| \nabla _h\phi \right\| _2^{1-\alpha },\alpha = \frac{4-D}{4},D=1,2,3\), which we establish for finite difference functions in an appendix. In Sect. 3 we define the fully discrete scheme and give some of its basic properties, including energy stability and solvability. In Sect. 4 we give the details of the convergence analyses, and in Sect. 5 we provide the results of some numerical tests that confirmed some of the theoretical predictions. We relegate many of the technical details that would slow the reading of the paper to three appendices.

2 The discrete periodic convolution and useful inequalities

Here we define a discrete periodic convolution operator on a 2D periodic grid. We need two spaces: \({\mathcal C}_{m\times n}\) is the space of cell-centered grid (or lattice) functions, and \(\mathcal V_{m\times n}\) is the space of vertex-centered grid functions. The precise definitions can be found in App. 6. The spaces and the following have straightforward extensions to three dimensions. Suppose \(\phi \in \mathcal C_{m\times n}\) is periodic and \(f \in \mathcal V_{m\times n}\) is periodic. Then the discrete convolution operator \(\left[ f \star \phi \right] : {\mathcal V}_{m \times n} \times {\mathcal C}_{m\times n} \rightarrow {\mathcal C}_{m\times n}\) is defined component-wise as

$$\begin{aligned} \left[ f \star \phi \right] _{i,j} := h^2\sum ^{m}_{k=1}\sum ^{n}_{l=1} f_{k+\frac{1}{2},l+\frac{1}{2}}\phi _{i-k,j-l}. \end{aligned}$$
(2.1)

Note very carefully that the order is important in the definition of the discrete convolution \(\left[ \, \cdot \, \star \, \cdot \, \right] \). The next result follows from the definition of the discrete convolution and simple estimates. The proof is omitted.

Lemma 2.1

If \(\phi , \psi \in \mathcal {C}_{m \times n}\) are periodic and \(f \in \mathcal {V}_{m \times n}\) is periodic and even, i.e., \(f_{i+\frac{1}{2},j+\frac{1}{2}} = f_{-i+\frac{1}{2},-j+\frac{1}{2}}\), for all \(i,j\in \mathbb {Z}\), then

$$\begin{aligned} \left( \phi \Vert \left[ f \star \psi \right] \right) = \left( \psi \Vert \left[ f \star \phi \right] \right) . \end{aligned}$$
(2.2)

If, in addition, \(f\) is non-negative then

$$\begin{aligned} \left| \left( \phi \Vert \left[ f \star \psi \right] \right) \right| \le \left[ f \star \mathbf{1} \right] \left( \frac{\alpha }{2} \left( \phi \Vert \phi \right) +\frac{1}{2\alpha }\left( \psi \Vert \psi \right) \right) , \quad \forall \alpha > 0. \end{aligned}$$
(2.3)

The detailed proofs of the next two results are presented in the appendix. The following lemma is the key to the \(\ell _2\) convergence analysis.

Lemma 2.2

Suppose \(\phi ,\, \psi \in \mathcal {C}_{m\times n}\) are periodic with equal means, i.e., \(\left( \phi -\psi \Vert \mathbf{1} \right) =0\). Suppose that \(\phi \) and \(\psi \) are in the classes of grid functions satisfying

$$\begin{aligned} \left\| \phi \right\| _4 + \left\| \phi \right\| _{\infty } + \left\| \nabla _h\phi \right\| _{\infty } \le C_{0}, \quad \left\| \psi \right\| _4 \le C_{0}, \end{aligned}$$
(2.4)

where \(C_{0}\) is an \(h\)-independent positive constant. Then, there exists a positive constant \(C_1\), which depends on \(C_0\) but is independent of \(h\), such that

$$\begin{aligned} 2h^2\left( \phi ^3 - \psi ^3 \Vert \Delta _h \left( \phi -\psi \right) \right) \le \frac{C_1}{\alpha ^3} \left\| \phi -\psi \right\| _2^2 + \alpha \left\| \nabla _h\left( \phi -\psi \right) \right\| _2^2 , \quad \forall \alpha > 0.\nonumber \\ \end{aligned}$$
(2.5)

Lemma 2.3

Suppose \(\phi , \psi \in \mathcal {C}_{m \times n}\) are periodic. Assume that \(\mathsf {f} \in C_\mathrm{per}^\infty (\Omega )\) is even and define its grid restriction via \(f_{i+\frac{1}{2},j+\frac{1}{2}} := \mathsf {f}\left( p_{i+\frac{1}{2}},p_{j+\frac{1}{2}}\right) \), so that \(f\in \mathcal {V}_{m\times n}\). Then for any \(\alpha > 0\), we have

$$\begin{aligned} - 2h^2\left( \left[ f \star \psi \right] \Vert \Delta _h \phi \right) \le \frac{C_2}{\alpha } \left\| \psi \right\| ^2_2 + \alpha \left\| \nabla _h \phi \right\| ^2_2, \end{aligned}$$
(2.6)

where \(C_2\) is a positive constant that depends on \(\mathsf {f}\) but is independent of \(h\).

3 A convex splitting scheme and its properties

3.1 Discrete energy and a fully discrete convex splitting scheme

We begin by defining a fully discrete energy that is consistent with the energy (1.2) in continuous space. In particular, the discrete energy \(F:{\mathcal C}_{m \times n}\rightarrow \mathbb {R}\) is given by

$$\begin{aligned} F(\phi ) := \frac{1}{4}\left\| \phi \right\| _4^4+\frac{\gamma _c-\gamma _e}{2} \left\| \phi \right\| _2^2 + \frac{\left[ J \star \mathbf{1} \right] }{2}\left\| \phi \right\| _2^2 - \frac{h^2}{2}\left( \phi \Vert \left[ J \star \phi \right] \right) , \end{aligned}$$
(3.1)

where \(J := J_c-J_e\), and \(J_\alpha \in \mathcal {V}_{m \times n}\), \(\alpha = c,e\), are defined via the vertex-centered grid restrictions \(\left( J_\alpha \right) _{i+\frac{1}{2},j+\frac{1}{2}} := \mathsf {J}_\alpha (p_{i+\frac{1}{2}},p_{j+\frac{1}{2}})\).

Lemma 3.1

Suppose that \(\phi \in \mathcal {C}_{m \times n}\) is periodic and define

$$\begin{aligned} F_c(\phi )&:= \frac{1}{4} \left\| \phi \right\| _4^4 +\frac{2\left[ J_c \star \mathbf{1} \right] +\gamma _c }{2} \left\| \phi \right\| _2^2,\end{aligned}$$
(3.2)
$$\begin{aligned} F_e(\phi )&:= \frac{ \left[ J_c \star 1 \right] +\left[ J_e \star \mathbf{1} \right] +\gamma _e }{2} \left\| \phi \right\| _2^2 + \frac{h^2}{2}\left( \phi \Vert \left[ J \star \phi \right] \right) . \end{aligned}$$
(3.3)

Then \(F_c\) and \(F_e\) are convex, and the gradients of the respective energies are

$$\begin{aligned} \delta _\phi F_c = \phi ^3 +\left( 2\left[ J_c \star \mathbf{1} \right] +\gamma _c\right) \phi \ , \quad \delta _\phi F_e = \left( \left[ J_c \star \mathbf{1} \right] +\left[ J_e \star \mathbf{1} \right] +\gamma _e\right) \phi + \left[ J \star \phi \right] .\nonumber \\ \end{aligned}$$
(3.4)

Hence \(F\), as defined in (3.1), admits the convex splitting \(F = F_c-F_e\).

Proof

\(F_c\) is clearly convex. To see that \(F_e\) is convex, make use of the estimate (2.3) and observe that \(\left. \frac{d^2}{ds^2} F_e(\phi +s\psi )\right| _{s=0} \ge 0\), for any periodic \(\psi \in \mathcal {C}_{m\times n}\). We suppress the details for brevity. \(\square \)

We now describe the fully discrete schemes in detail. Define the cell-centered chemical potential \(w\in {\mathcal C}_{m \times n}\) to be

$$\begin{aligned} w^{k+1} := w\left( \phi ^{k+1},\phi ^k\right)&:= \delta _\phi F_c(\phi ^{k+1}) -\delta _\phi F_e(\phi ^k) \nonumber \\&= \left( \phi ^{k+1} \right) ^3+\left( 2\left[ J_c \star \mathbf{1} \right] +\gamma _c\right) \phi ^{k+1} \nonumber \\&- \left( \left[ J_c \star \mathbf{1} \right] +\left[ J_e \star \mathbf{1} \right] +\gamma _e\right) \phi ^k - \left[ J \star \phi ^k \right] .\quad \quad \end{aligned}$$
(3.5)

The scheme for conserved dynamics can be formulated as: given \(\phi ^k\in {\mathcal C}_{m \times n}\) periodic, find \(\phi ^{k+1},\, w^{k+1} \in {\mathcal C}_{m \times n}\) periodic so that

$$\begin{aligned} \phi ^{k+1}-\phi ^k = s \Delta _h w^{k+1} , \end{aligned}$$
(3.6)

where \(\Delta _h\) is the standard five-point discrete laplacian operator.

3.2 Unconditional solvability and energy stability

Now we show that the convex splitting framework automatically confers unconditional solvability and stability properties to our scheme (3.6). The solvability follows immediately from the fact that \(F_c\) is convex. The details for the proof were established in [39]—see also  [38, 40]—and we therefore omit them for brevity.

Theorem 3.2

The scheme (3.6) is discretely mass conservative, i.e., \(\left( \phi ^{k+1}-\phi ^k \Vert \mathbf{1} \right) =0\), for all \(k\ge 0\), and uniquely solvable for any time step size \(s>0\).

Lemma 3.3

Suppose that \(\phi \in {\mathcal C}_{m\times n}\) is periodic and the discrete energy \(F\) is as defined as Eq. (3.1). There exists a non-negative constant \(C\) such that \(F(\phi ) +C \ge 0\). More specifically,

$$\begin{aligned} \frac{1}{8}\left\| \phi \right\| _{4}^4&\le F(\phi ) + \frac{\left( \gamma _c-\gamma _e-2\left[ J_e \star \mathbf{1} \right] \right) ^2}{2}|\Omega |,\end{aligned}$$
(3.7)
$$\begin{aligned} \frac{1}{2}\left\| \phi \right\| _{2}^2&\le F(\phi ) +\frac{\left( \gamma _c-\gamma _e-2\left[ J_e \star \mathbf{1} \right] -1\right) ^2}{4}|\Omega |. \end{aligned}$$
(3.8)

Proof

Using the estimate (2.3),

$$\begin{aligned} \frac{\left[ J \star \mathbf{1} \right] }{2}\left\| \phi \right\| _{2}^2 -\frac{h^2}{2}\left( \phi \Vert \left[ J \star \phi \right] \right)&= \frac{\left[ J_c \star \mathbf{1} \right] }{2}\left\| \phi \right\| _{2}^2 -\frac{h^2}{2}\left( \phi \Vert \left[ J_c \star \phi \right] \right) \nonumber \\&- \frac{\left[ J_e \star \mathbf{1} \right] }{2}\left\| \phi \right\| _{2}^2 +\frac{h^2}{2}\left( \phi \Vert \left[ J_e \star \phi \right] \right) \nonumber \\&\ge - \frac{\left[ J_e \star \mathbf{1} \right] }{2}\left\| \phi \right\| _{2}^2 +\frac{h^2}{2}\left( \phi \Vert \left[ J_e \star \phi \right] \right) \nonumber \\&\ge - \frac{\left[ J_e \star \mathbf{1} \right] }{2}\left\| \phi \right\| _{2}^2 - \frac{\left[ J_e \star \mathbf{1} \right] }{2}\left\| \phi \right\| _{2}^2 \nonumber \\&= - \left[ J_e \star \mathbf{1} \right] \left\| \phi \right\| _{2}^2 \end{aligned}$$
(3.9)

Consider the polynomials

$$\begin{aligned} P(t) := \frac{1}{4} t^4 + \frac{\gamma _c}{2} t^2-\frac{\gamma _e}{2} t^2 \ , \quad Q(t) :=P( t)-\left[ J_e \star \mathbf{1} \right] t^2. \end{aligned}$$
(3.10)

Observe that

$$\begin{aligned} F(\phi ) = h^2\left( P(\phi ) \Vert \mathbf{1} \right) + \frac{\left[ J \star \mathbf{1} \right] }{2}\left\| \phi \right\| _{2}^2 -\frac{h^2}{2}\left( \phi \Vert \left[ J \star \phi \right] \right) \end{aligned}$$
(3.11)

and, using (3.9),

$$\begin{aligned} F(\phi ) \ge h^2\left( Q(\phi ) \Vert \mathbf{1} \right) . \end{aligned}$$
(3.12)

Utilizing the polynomial inequalities

$$\begin{aligned} \begin{aligned} \frac{1}{8} t^4&\le Q(t) +\frac{\left( \gamma _c-\gamma _e-2\left[ J_e \star \mathbf{1} \right] \right) ^2}{2} , \\ \frac{1}{2} t^2&\le Q(t) +\frac{\left( \gamma _c-\gamma _e-2\left[ J_e \star 1 \right] -1\right) ^2}{4} \ , \end{aligned} \end{aligned}$$
(3.13)

the estimates follow. \(\square \)

The following is a discrete version of Theorem 1.2.

Theorem 3.4

Suppose the energy \(F(\phi )\) is as defined in Eq. (3.1), assume \(\phi ^{k+1},\, \phi ^k\in {\mathcal C}_{m \times n}\) are periodic and they are solutions to the scheme (3.6). Then for any \(s>0\)

$$\begin{aligned} F\left( \phi ^{k+1}\right) + s\left\| \nabla _h w^{k+1} \right\| _{2}^2 + R_h\left( \phi ^k,\phi ^{k+1}\right) = F\left( \phi ^k\right) , \end{aligned}$$
(3.14)

with the remainder term, \(R_h\), given by

$$\begin{aligned} R_h\left( \phi ^k,\phi ^{k+1}\right)&= \frac{1}{4}\left\| \left( \phi ^{k+1}\right) ^2-\left( \phi ^k\right) ^2 \right\| _{2}^2 + \frac{1}{2}\left\| \phi ^{k+1}\left( \phi ^{k+1}-\phi ^k\right) \right\| _{2}^2 \nonumber \\&+\frac{3\left[ J_c \star \mathbf{1} \right] + \left[ J_e \star \mathbf{1} \right] +\gamma _c+\gamma _e}{2}\left\| \phi ^{k+1}-\phi ^k \right\| _{2}^2 \nonumber \\&+ \frac{h^2}{2}\left( {\left[ J \star \left( \phi ^{k+1}-\phi ^k\right) \right] }\Big \Vert {\phi ^{k+1}-\phi ^k}\right) . \end{aligned}$$
(3.15)

Most importantly, \(R_h\left( \phi ^k,\phi ^{k+1}\right) \) is non-negative, which implies that the energy is non-increasing for any \(s\), i.e., \(F\left( \phi ^{k+1}\right) \le F\left( \phi ^k\right) \).

Proof

To obtain (3.14), we note the following identities. First, for the quartic terms,

$$\begin{aligned} h^2\left( {\left( \phi ^{k+1}\right) ^3}\Big \Vert {\phi ^{k+1}-\phi ^k}\right)&= \frac{1}{4}\left\| \phi ^{k+1} \right\| _{4}^4 - \frac{1}{4}\left\| \phi ^k \right\| _{4}^4 + \frac{1}{4}\left\| \left( \phi ^{k+1}\right) ^2-\left( \phi ^k\right) ^2 \right\| _{2}^2 \nonumber \\&+ \frac{1}{2}\left\| \phi ^{k+1}\left( \phi ^{k+1}-\phi ^k\right) \right\| _{2}^2. \end{aligned}$$
(3.16)

Likewise

$$\begin{aligned} h^2\left( {\phi ^{k+1}}\Big \Vert {\phi ^{k+1}-\phi ^k}\right) = \frac{1}{2}\left\| \phi ^{k+1} \right\| _{2}^2 -\frac{1}{2}\left\| \phi ^k \right\| _{2}^2 +\frac{1}{2}\left\| \phi ^{k+1}-\phi ^k \right\| _{2}^2 ,\quad \quad \end{aligned}$$
(3.17)

and

$$\begin{aligned} -h^2\left( {\phi ^k}\Big \Vert {\phi ^{k+1}-\phi ^k}\right) = \frac{1}{2}\left\| \phi ^k \right\| _{2}^2 -\frac{1}{2}\left\| \phi ^{k+1} \right\| _{2}^2 +\frac{1}{2}\left\| \phi ^{k+1}-\phi ^k \right\| _{2}^2.\quad \quad \end{aligned}$$
(3.18)

Making use of (2.2), we find

$$\begin{aligned} -h^2\left( {\left[ J \star \phi ^k \right] }\Big \Vert {\phi ^{k+1}-\phi ^k}\right)&= \frac{h^2}{2}\left( {\left[ J \star \phi ^k \right] }\Big \Vert {\phi ^k}\right) - \frac{h^2}{2}\left( {\left[ J \star \phi ^{k+1} \right] }\Big \Vert {\phi ^{k+1}}\right) \nonumber \\&+ \frac{h^2}{2}\left( {\left[ J \star \left( \phi ^{k+1}-\phi ^k\right) \right] }\Big \Vert {\phi ^{k+1}-\phi ^k}\right) .\quad \quad \quad \quad \quad \end{aligned}$$
(3.19)

Now, testing Eq. (3.6) with \(w^{k+1}\), we obtain

$$\begin{aligned} h^2\left( {\phi ^{k+1}-\phi ^k}\Big \Vert {w^{k+1}}\right) = -s\left\| \nabla _h w^{k+1} \right\| _{2}^2, \end{aligned}$$
(3.20)

where

$$\begin{aligned} \left( \phi ^{k+1}-\phi ^k \Vert w^{k+1} \right) \!&= \! \left( {\left( \phi ^{k+1}\right) ^3}\Big \Vert {\phi ^{k+1}\!-\!\phi ^k}\right) \!+\! \left( 2\left[ J_c \star \mathbf{1} \right] \!+\!\gamma _c\right) \left( {\phi ^{k+1}}\Big \Vert {\phi ^{k+1}\!-\!\phi ^k}\right) \nonumber \\&- \left( \left[ J_c \star \mathbf{1} \right] +\left[ J_e \star \mathbf{1} \right] +\gamma _e\right) \left( {\phi ^k}\Big \Vert {\phi ^{k+1}-\phi ^k}\right) \nonumber \\&- \left( {\left[ J \star \phi ^k \right] }\Big \Vert {\phi ^{k+1}-\phi ^k}\right) . \end{aligned}$$
(3.21)

Using identities (3.16)–(3.19) in the last equation, we get

$$\begin{aligned} h^2\left( {\phi ^{k+1}-\phi ^k}\Big \Vert {w^{k+1}}\right) = F\left( \phi ^{k+1}\right) - F\left( \phi ^k\right) +R_h\left( \phi ^k,\phi ^{k+1}\right) . \end{aligned}$$
(3.22)

Next, we show that \(R_h\left( \phi ^k,\phi ^{k+1}\right) \ge 0\). Clearly

$$\begin{aligned} R_h\left( \phi ^k,\phi ^{k+1}\right)&\ge \frac{3\left[ J_c \star \mathbf{1} \right] + \left[ J_e \star \mathbf{1} \right] }{2}\left\| \phi ^{k+1}-\phi ^k \right\| _{2}^2 \nonumber \\&+ \frac{h^2}{2}\left( {\left[ J \star \left( \phi ^{k+1}-\phi ^k\right) \right] }\Big \Vert {\phi ^{k+1}-\phi ^k}\right) \nonumber \\&= \frac{3\left[ J_c \star \mathbf{1} \right] }{2}\left\| \phi ^{k+1}-\phi ^k \right\| _{2}^2 + \frac{\left[ J_e \star \mathbf{1} \right] }{2}\left\| \phi ^{k+1}-\phi ^k \right\| _{2}^2 \nonumber \\&+ \frac{h^2}{2}\left( {\left[ J_c \star \left( \phi ^{k+1}-\phi ^k\right) \right] }\Big \Vert {\phi ^{k+1}-\phi ^k}\right) \nonumber \\&-\frac{h^2}{2}\left( {\left[ J_e \star \left( \phi ^{k+1}-\phi ^k\right) \right] }\Big \Vert {\phi ^{k+1}-\phi ^k}\right) \nonumber \\&\ge \left[ J_c \star \mathbf{1} \right] \left\| \phi ^{k+1}-\phi ^k \right\| _{2}^2 \ , \end{aligned}$$
(3.23)

where we have used estimate (2.3) twice in the last step. The result is proved. \(\square \)

Remark 3.5

If we are not interested in the specific form of \(R_h\), only that \(R_h\ge 0\), then the previous result, the unconditional energy stability, follows immediately from the fact that we use a convex splitting scheme, appealing to [39, Thms. 3.5 and 3.6].

Putting Lemma 3.3 and Theorem 3.4 together, we immediately get the following.

Corollary 3.6

Suppose that \(\left\{ \phi ^k,w^k\right\} _{k=1}^l\in \left[ {\mathcal C}_{m \times n}\right] ^2\) are a sequence of periodic solution pairs of the scheme (3.6) with the starting values \(\phi ^0\). Then, for any \(1 \le k \le l\),

$$\begin{aligned} \frac{1}{8}\left\| \phi ^k \right\| _{4}^4&\le F\left( \phi ^0\right) + \frac{\left( \gamma _c-\gamma _e-2\left[ J_e \star \mathbf{1} \right] \right) ^2}{2}|\Omega | ,\end{aligned}$$
(3.24)
$$\begin{aligned} \frac{1}{2}\left\| \phi ^k \right\| _{2}^2&\le F\left( \phi ^0\right) +\frac{\left( \gamma _c-\gamma _e-2\left[ J_e \star \mathbf{1} \right] -1\right) ^2}{4}|\Omega |. \end{aligned}$$
(3.25)

Theorem 3.7

Suppose \(\Phi \in C_\mathrm{per}^\infty (\Omega )\) and set \(\phi ^0_{i,j} := \Phi (p_i,p_j)\). Suppose that \(\left\{ \phi ^k,w^k\right\} _{k=1}^l\in \left[ {\mathcal C}_{m \times n}\right] ^2\) are a sequence of periodic solution pairs of the scheme (3.6) with the starting values \(\phi ^0\). There exist constants \(C_3,\, C_4,\, C_5 >0\), which are independent of \(h\) and \(s\), such that

$$\begin{aligned} \max _{1\le k\le l}\left\| \phi ^k \right\| _{4}&\le C_3.\end{aligned}$$
(3.26)
$$\begin{aligned} \max _{1\le k\le l}\left\| \phi ^k \right\| _2&\le C_4 , \end{aligned}$$
(3.27)

and

$$\begin{aligned} \sqrt{s\sum _{k = 1}^\ell \left\| \nabla _h w^{k+1} \right\| _{2}^2 \ } \le C_5. \end{aligned}$$
(3.28)

Proof

Since the discrete energy \(F\) is consistent with the continuous energy \(E\),

$$\begin{aligned} F\left( \phi ^0\right) \le E\left( \Phi \right) + C h^2 \le E\left( \Phi \right) + C |\Omega | , \end{aligned}$$
(3.29)

where \(C>0\) is independent of \(h\). Invoking Cor. 3.6 and using a second consistency argument on the discrete convolution \(\left[ J_e \star \mathbf{1} \right] \), for all \(1\le k\le l\), we have

$$\begin{aligned} \frac{1}{8}\left\| \phi ^k \right\| _{4}^4&\le F\left( \phi ^0\right) + \frac{\left( \gamma _c-\gamma _e-2\left[ J_e \star \mathbf{1} \right] \right) ^2}{2}|\Omega | ,\nonumber \\&\le E\left( \Phi \right) + \frac{\left( \gamma _c-\gamma _e-2\left( J_e*1\right) \right) ^2}{2}|\Omega | +C|\Omega |. \end{aligned}$$
(3.30)

The right-hand-side is clearly independent of \(h\) and \(s\), and (3.26) follows. Estimate (3.27) is similar. Finally estimate (3.28) follows by summing (3.14) and proceeding as above. \(\square \)

4 \(\ell ^\infty \left( \ell ^2\right) \) and \(\ell ^{\infty }\left( \ell ^\infty \right) \) norm error estimates

We conclude this section with two local-in-time error estimates for our convex splitting scheme (3.6) in two dimensions. The extension of the proofs to three dimensions is omitted for the sake of brevity, but see Remark 7.5 for some of the details. First, we establish an error estimate in the \(\ell ^\infty (0,T; \ell ^2)\) norm that has no restriction on the refinement path. Then we prove an estimate in the \(\ell ^\infty (0,T; \ell ^\infty )\) norm under the mild linear refinement path constraint \(s \le C h\).

The existence and uniqueness of a smooth, periodic solution to the IPDE (1.9) with smooth periodic initial data may be established using techniques developed by Bates and Han in [8, 9]. In the following pages we denote this IPDE solution by \(\Phi \). Motivated by the results of Bates and Han, and based on the assumptions in the introduction, it will be reasonable to assume that

$$\begin{aligned} \left\| \Phi \right\| _{L^\infty (0,T;L^4)} + \left\| \Phi \right\| _{L^\infty (0,T;L^\infty )} + \left\| \nabla \Phi \right\| _{L^\infty (0,T;L^\infty )} < C , \end{aligned}$$
(4.1)

for any \(T >0\), and therefore also, using a consistency argument, that

$$\begin{aligned} \max _{1\le k\le \ell } \left\| \Phi ^k \right\| _4 + \max _{1\le k\le \ell }\left\| \Phi ^k \right\| _\infty + \max _{1\le k\le \ell }\left\| \nabla _h \Phi ^k \right\| _\infty < C , \end{aligned}$$
(4.2)

after setting \(\Phi ^k_{i,j} := \Phi (p_i,p_j,t_k)\), where \(C\) is independent of \(h\) and \(s\) and \(t_k=k \cdot s\). The IPDE solution \(\Phi \) is also mass conservative, meaning that, for all \(0\le t\le T,\int _\Omega \left( \Phi (\mathbf{x},0) - \Phi (\mathbf{x},t)\right) d\mathbf{x}= 0\). For our numerical scheme, on choosing \(\phi _{i,j}^0 := \Phi _{i,j}^0\), we note that \(\left( \phi ^k-\Phi ^0 \Vert \mathbf{1} \right) = 0\), for all \(k\ge 0\). But, unfortunately, \(\left( \phi ^k-\Phi ^k \Vert \mathbf{1} \right) \ne 0\) in general. On the other hand, by consistency,

$$\begin{aligned} \frac{1}{L_1L_2}h^2\left( {\phi ^k-\Phi ^k}\Big \Vert {\mathbf{1}}\right) =: \beta ^k, \quad \left| \beta ^k\right| \le Ch^2, \end{aligned}$$
(4.3)

for some \(C>0\) that is independent of \(k\) and \(h\), for all \(1\le k \le l\). We set \(\tilde{\Phi }^k := \Phi ^k+\beta ^k\) and observe \(\big ({\phi ^k-\Phi ^k}\big \Vert {\mathbf{1}}\big ) = 0\) and also

$$\begin{aligned} \max _{1\le k\le \ell } \left\| \tilde{\Phi }^k \right\| _4 + \max _{1\le k\le \ell }\left\| \tilde{\Phi }^k \right\| _\infty + \max _{1\le k\le \ell }\left\| \nabla _h \tilde{\Phi }^k \right\| _\infty < C. \end{aligned}$$
(4.4)

Finally, the assumptions on the continuous kernel \(\mathsf {J}\), specifically (1.12), and the consistency of the discrete convolution will further imply

$$\begin{aligned} \left[ J \star \mathbf{1} \right] + \gamma _c - \gamma _e = \alpha _0 > 0, \end{aligned}$$
(4.5)

for some \(\alpha _0\) that is independent of \(h\), provided \(h\) is sufficiently small.

Theorem 4.1

Given smooth, periodic initial data \(\Phi (x_1,x_2,t=0)\), suppose the unique, smooth, periodic solution for the IPDE (1.9) is given by \(\Phi (x,y,t)\) on \(\Omega \) for \(0<t\le T\), for some \(T< \infty \). Define \(\Phi _{i,j}^k\) as above and set \(e_{i,j}^k := \Phi _{i,j}^k-\phi _{i,j}^k\), where \(\phi ^k_{i,j}\in {\mathcal C}_{m \times n}\) is \(k^\mathrm{th}\) periodic solution of (3.6) with \(\phi ^0_{i,j} := \Phi ^0_{i,j}\). Then, provided \(s\) and \(h\) are sufficiently small, for all positive integers \(l\), such that \(l\, s \le T\), we have

$$\begin{aligned} \left\| e^l \right\| _2 \le C\left( h^2+s\right) , \end{aligned}$$
(4.6)

where \(C>0\) is independent of \(h\) and \(s\).

Proof

By consistency, the IPDE solution \(\Phi \) solves the discrete equation

$$\begin{aligned} \Phi ^{k+1} - \Phi ^{k} = s\Delta _hw\left( \Phi ^{k+1},\Phi ^{k}\right) +s\tau ^{k+1} , \end{aligned}$$
(4.7)

where the local truncation error \(\tau ^{k+1}\) satisfies

$$\begin{aligned} \left| \tau _{i,j}^{k+1}\right| \le C_6 \left( h^2 + s\right) , \end{aligned}$$
(4.8)

for all \(i\), \(j\), and \(k\) for some \(C_6\ge 0\) that depends only on \(T\), \(L_1\) and \(L_2\). Subtracting (3.6) from (4.7) yields

$$\begin{aligned} e^{k+1} - e^k = s\Delta _h\left( w\left( \Phi ^{k+1},\Phi ^k\right) - w\left( \phi ^{k+1},\phi ^k\right) \right) + s\tau ^{k+1}. \end{aligned}$$
(4.9)

Multiplying by \(2h^2e^{k+1}\), summing over \(i\) and \(j\), and applying Green’s second identity (6.3) we have

$$\begin{aligned} \left\| e^{k+1} \right\| _2^2 \!-\! \left\| e^k \right\| _2^2 \!+\! \left\| e^{k+1}-e^k \right\| _2^2 \!&= \! 2 h^2 s \left( w\left( \Phi ^{k+1},\Phi ^k\right) \!-\! w\left( \phi ^{k+1},\phi ^k\right) \Big \Vert \Delta _he^{k+1} \right) \nonumber \\&+\,s2h^2 \left( \tau ^{k+1} \Big \Vert e^{k+1} \right) \nonumber \\&= 2sh^2 \left( \Delta _h\left( \left( \Phi ^{k+1}\right) ^3-\left( \phi ^{k+1} \right) ^3\right) \Big \Vert e^{k+1} \right) \nonumber \\&+\, 2s h^2 \left( 2\left[ J_c \star \mathbf{1} \right] +\gamma _c\right) \left( e^{k+1} \Big \Vert \Delta _he^{k+1} \right) \nonumber \\&-\, 2 sh^2 \left( \left[ J_c \star \mathbf{1} \right] +\left[ J_e \star \mathbf{1} \right] \!+\!\gamma _e\right) \left( e^{k} \Big \Vert \Delta _he^{k+1} \right) \nonumber \\&-\, 2 h^2s\left( \left[ J \star e^{k} \right] \Big \Vert \Delta _he^{k+1} \right) \nonumber \\&+\, 2sh^2 \left( \tau ^{k+1} \Big \Vert e^{k+1} \right) . \end{aligned}$$
(4.10)

To estimate the nonlinear term we need to work with \(\tilde{\Phi }^k:= \Phi ^k +\beta ^k\) rather than \(\Phi ^k\), because we require \(\big ({\phi ^k-\tilde{\Phi }^k}\big \Vert {\mathbf{1}}\big ) = 0\) to use Lemma 2.2. The switch gives another consistency error, \(\tilde{\tau }^k\), of the same order as \(\tau ^k\):

$$\begin{aligned} \left\| e^{k+1} \right\| _2^2 - \left\| e^k \right\| _2^2 + \left\| e^{k+1}-e^k \right\| _2^2&= 2sh^2 \left( \left( \tilde{\Phi }^{k+1} \right) ^3-\left( \phi ^{k+1} \right) ^3 \Big \Vert \Delta _he^{k+1} \right) \nonumber \\&+\, 2s h^2 \left( 2\left[ J_c \star \mathbf{1} \right] +\gamma _c\right) \left( e^{k+1} \Big \Vert \Delta _he^{k+1} \right) \nonumber \\&-\, 2 sh^2 \left( \left[ J_c \star \mathbf{1} \right] \!+\!\left[ J_e \star \mathbf{1} \right] \!+\!\gamma _e\right) \left( e^{k} \Big \Vert \Delta _he^{k+1} \right) \nonumber \\&-\, 2 h^2s\left( \left[ J \star e^{k} \right] \Big \Vert \Delta _he^{k+1} \right) \nonumber \\&+\, 2sh^2 \left( \tau ^{k+1}+\tilde{\tau }^{k+1} \Big \Vert e^{k+1} \right) . \end{aligned}$$
(4.11)

We now make a key estimate. Set \(\tilde{e}^k:=\tilde{\Phi }^k-\phi ^k\), and observe that \(\left( \tilde{e}^k \Vert \mathbf{1} \right) = 0\), \(1\le k\le l\). Applying Lemma 2.2 we obtain

$$\begin{aligned} 2h^2s\left( \left( \tilde{\Phi }^{k+1}\right) ^3 - \left( \phi ^{k+1}\right) ^3 \Big \Vert \Delta _he^{k+1} \right)&= 2h^2s\left( \left( \tilde{\Phi }^{k+1}\right) ^3 - \left( \phi ^{k+1}\right) ^3 \Big \Vert \Delta _h\tilde{e}^{k+1} \right) \nonumber \\&\le \frac{C_1}{\alpha _0^3} s \left\| \tilde{e}^{k+1} \right\| _2^2 + \alpha _0 s \left\| \nabla _h \tilde{e}^{k+1} \right\| _2^2 \nonumber \\&\le \frac{2C_1}{\alpha _0^3} s \left\| e^{k+1} \right\| _2^2 \!+\!\frac{Csh^4}{\alpha _0^3} \!+ \! \alpha _0 s \left\| \nabla _h e^{k+1} \right\| _2^2,\nonumber \\ \end{aligned}$$
(4.12)

where we have taken \(\alpha = \alpha _0\) and \(\alpha _0\) is from (4.5). Note that \(C_1\) is independent of both \(s\) and \(h\). Using summation-by-parts, we get

$$\begin{aligned} 2 h^2s \left( 2\left[ J_c \star \mathbf{1} \right] +\gamma _c\right) \left( e^{k+1} \Big \Vert \Delta _he^{k+1} \right) = -2s \left( 2\left[ J_c \star \mathbf{1} \right] +\gamma _c\right) \left\| \nabla _h e^{k+1} \right\| _2^2.\quad \quad \quad \end{aligned}$$
(4.13)

Using summation-by-parts, the Cauchy-Schwartz inequality, and Cauchy’s inequality, we have

$$\begin{aligned}&-2 h^2s \left( \left[ J_c \star \mathbf{1} \right] +\left[ J_e \star \mathbf{1} \right] +\gamma _e\right) \left( e^{k} \Big \Vert \Delta _he^{k+1} \right) \nonumber \\&\quad \le s \left( \left[ J_c \star \mathbf{1} \right] +\left[ J_e \star \mathbf{1} \right] +\gamma _e\right) \left( \left\| \nabla _h e^{k} \right\| _{2}^2 + \left\| \nabla _h e^{k+1} \right\| _{2}^2\right) . \end{aligned}$$
(4.14)

By Lemma 2.3,

$$\begin{aligned} - 2 h^2s\left( \left[ J \star e^{k} \right] \Big \Vert \Delta _he^{k+1} \right) \le sC_2\frac{1}{\alpha } \left\| e^{k} \right\| ^2_2 + s \alpha \left\| \nabla _h e^{k+1} \right\| ^2_2 , \end{aligned}$$
(4.15)

for any \(\alpha >0\). Cauchy’s inequality shows that

$$\begin{aligned} 2h^2s\left( \tau ^{k+1}+\tilde{\tau }^{k+1} \Big \Vert e^{k+1} \right)&\le s C_7\left( h^2 +s \right) ^2 +s \left\| e^{k+1} \right\| _2^2. \end{aligned}$$
(4.16)

for some \(C_7 >0\). Combining the above results, we have

$$\begin{aligned}&\left\| e^{k+1} \right\| _{2}^2-\left\| e^{k} \right\| _{2}^2\nonumber \\&\quad \le \frac{C_1}{\alpha _0^3} s \left\| e^{k+1} \right\| ^2_{2} + \alpha _0 s \left\| \nabla _h e^{k+1} \right\| ^2_{2} - 2s \left( 2\left[ J_c \star \mathbf{1} \right] +\gamma _c\right) \left\| \nabla _h e^{k+1} \right\| ^2_2 \nonumber \\&\quad \quad +\, s\left( \left[ J_c \star \mathbf{1} \right] \!+\!\left[ J_e \star \mathbf{1} \right] \!+\!\gamma _e\right) \left\| \nabla _h e^{k} \right\| _{2}^2 \!+\! s\left( \left[ J_c \star \mathbf{1} \right] +\left[ J_e \star \mathbf{1} \right] +\gamma _e\right) \left\| \nabla _h e^{k+1} \right\| _{2}^2 \nonumber \\&\quad \quad +\, s\frac{C_2}{\alpha } \left\| e^{k} \right\| ^2_2 + s\alpha \left\| \nabla _h e^{k+1} \right\| ^2_2 + s C_7 (h^2 + s)^2 +\frac{Csh^4}{\alpha _0^3} + s\left\| e^{k+1} \right\| _{2}^2.\quad \quad \quad \quad \end{aligned}$$
(4.17)

Therefore,

$$\begin{aligned}&\left\| e^{k+1} \right\| _{2}^2-\left\| e^{k} \right\| _{2}^2 + s\left( 3 \left[ J_c \star \mathbf{1} \right] - \left[ J_e \star \mathbf{1} \right] + 2 \gamma _c - \gamma _e - \alpha _0 - \alpha \right) \left\| \nabla _h e^{k+1} \right\| _{2}^2 \nonumber \\&\quad \le s\left( \left[ J_c \star \mathbf{1} \right] +\left[ J_e \star \mathbf{1} \right] +\gamma _e \right) \left\| \nabla _h e^k \right\| _{2}^2 + s \left( C_7+\frac{C}{\alpha _0^3}\right) (h^2 + s)^2 \nonumber \\&\quad \quad + \frac{C_2}{\alpha } \left\| e^{k} \right\| ^2_2 + s \left( 1 + \frac{C_1}{\alpha _0^3} \right) \left\| e^{k+1} \right\| _{2}^2. \end{aligned}$$
(4.18)

Summing on \(k\), from \(k=0\) to \(k = l-1\), and using \(e^{0}\equiv 0\), we have

$$\begin{aligned}&\left\| e^l \right\| _{2}^2 + s \left( \alpha _0 - \alpha \right) \sum _{k=1}^{l-1} \left\| \nabla _h e^{k} \right\| _{2}^2 +s\left( 2\left[ J_c \star \mathbf{1} \right] +\gamma _c-\alpha \right) \left\| \nabla _h e^l \right\| _{2}^2 \nonumber \\&\quad \le s\frac{C_2}{\alpha } \sum _{k=1}^{l-1}\left\| e^{k} \right\| _{2}^2 + s \left( 1 + \frac{C_1}{\alpha _0^3} \right) \sum _{k=0}^{l-1}\left\| e^{k+1} \right\| _{2}^2 +s\left( C_7+\frac{C}{\alpha _0^3}\right) \sum _{k=0}^{l-1}(h^2 + s)^2 ,\nonumber \\ \end{aligned}$$
(4.19)

where we have used \(\left[ J_c \star \mathbf{1} \right] -\left[ J_e \star \mathbf{1} \right] + \gamma _c - \gamma _e = \alpha _0\) as in (4.5). Assuming that \(h\) is small enough so that the inequality in (4.5) holds and taking \(\alpha = \frac{\alpha _0}{2}\), we obtain

$$\begin{aligned} \alpha _0 - \alpha = \frac{\alpha _0}{2} > 0, \quad \! 2\left[ J_c \star \mathbf{1} \right] +\gamma _c-\alpha = \frac{3}{2}\left[ J_c \star \mathbf{1} \right] \!+\!\frac{1}{2}\left[ J_e \star \mathbf{1} \right] \!+\! \frac{1}{2}\gamma _c\!+\! \frac{1}{2}\gamma _e \!\ge \! 0.\nonumber \\ \end{aligned}$$
(4.20)

As a direct consequence, the following inequality holds:

$$\begin{aligned} \left\| e^l \right\| _{2}^2 \le s\frac{2C_2}{\alpha _0} \sum _{k=1}^{l-1}\left\| e^{k} \right\| _{2}^2 + s C_8 \sum _{k=1}^l\left\| e^k \right\| _{2}^2 +s C_9 \sum _{k=0}^{l-1}(h^2 + s)^2 , \end{aligned}$$
(4.21)

with \(C_8 := 1 + \frac{C_1}{\alpha _0^3}\) and \(C_9 := C_7+\frac{C}{\alpha _0^3}\). Simplifying,

$$\begin{aligned} \left\| e^l \right\| _{2}^2 \le s C_8 \left\| e^l \right\| _{2}^2 + s\left( \frac{2C_2}{\alpha _0} + C_8\right) \sum _{k=1}^{l-1}\left\| e^{k} \right\| _{2}^2+s C_9 \sum _{k=0}^{l-1}(h^2 + s)^2.\quad \quad \end{aligned}$$
(4.22)

Assuming \( s < \frac{1}{C_8}\) and \(l\, s \le T\), we arrive at

$$\begin{aligned} \left\| e^l \right\| _{2}^2 \le&\frac{\frac{2C_2}{\alpha _0} +C_8 }{1- C_8 s} s\sum \limits _{k=1}^{l-1}\left\| e^{k} \right\| _{2}^2 +\frac{TC_9}{1-C_8s} (h^2 + s)^2. \end{aligned}$$
(4.23)

An application of the discrete Gronwall inequality yields the desired result, and the proof is complete. \(\square \)

Using a higher order consistency analysis via a perturbation argument, we can use the last theorem to show an optimal order convergence of the method in the \(\ell ^\infty (0,T;\ell ^\infty )\) norm.

Theorem 4.2

Under the assumptions of Theorem 4.1 and enforcing the linear refinement path constraint \(s \le C h\), with \(C\) any fixed constant, we also have optimal order convergence of the numerical solution of the scheme (3.6) in the \(\ell ^\infty \) norm. Namely, if \(s\) and \(h\) are sufficiently small, for all positive integers \(l\), such that \(l\, s \le T\), we have

$$\begin{aligned} \left\| e^l \right\| _\infty \le C\left( h^2+s\right) , \end{aligned}$$
(4.24)

where \(C>0\) is independent of \(h\) and \(s\).

Proof

The local truncation error in (4.7) will not be enough to recover a full \(O (s + h^2)\) error estimate in the norm \(\left\| \, \cdot \, \right\| _\infty \). To remedy this, we need to construct supplementary fields, \(\Phi ^1_h,\Phi ^1_s\), and \(\hat{\Phi }\), satisfying

$$\begin{aligned} \hat{\Phi } = \Phi + h^2 \Phi ^1_{h} + s \Phi ^1_s , \end{aligned}$$
(4.25)

so that a higher \(O (s^2 + h^4)\) consistency is satisfied with the given numerical scheme (3.6). The constructed fields \(\Phi ^1_h,\Phi ^1_s\), which will be found using a perturbation expansion, will depend solely on the exact solution \(\Phi \).

The following truncation error analysis for the spatial discretization can be obtained by using a straightforward Taylor expansion for the exact solution:

$$\begin{aligned} \partial _t \Phi = \Delta _h \left( \Phi ^3 + \left( \left[ J \star \mathbf{1} \right] + \gamma _c - \gamma _e \right) \Phi - \left[ J \star \Phi \right] \right) + h^{2} {{\varvec{g}}}^{(0)} + O(h^{4}) , \quad \forall \ (i,j).\nonumber \\ \end{aligned}$$
(4.26)

Here the spatially discrete function \({{\varvec{g}}}^{(0)}\) is smooth enough in the sense that its discrete derivatives are bounded. Also note that there is no \(O(h^3)\) truncation error term, due to the fact that the centered difference used in the spatial discretization gives local truncation errors with only even order terms, \(O(h^2),O(h^4)\), et cetera.

The spatial correction function \(\Phi ^1_h\) is given by solving the following equation:

$$\begin{aligned} \partial _t \Phi ^1_h = \Delta _h \left( 3 \Phi ^2 \Phi ^1_h + \left( \left[ J \star \mathbf{1} \right] + \gamma _c - \gamma _e\right) \Phi ^1_h - \left[ J \star \Phi ^1_h \right] \right) - {{\varvec{g}}}^{(0)} , \quad \forall \ (i,j).\nonumber \\ \end{aligned}$$
(4.27)

Existence of a solution of the above linear system of ODEs is a standard exercise. Note that the solution depends only on the exact solution, \(\Phi \). In addition, the divided differences of \(\Phi ^1_h\) of various orders are bounded.

Now, we define

$$\begin{aligned} \Phi ^*_h := \Phi + h^2 \Phi ^1_{h}. \end{aligned}$$
(4.28)

A combination of (4.26) and (4.27) leads to the fourth order local truncation error for \(\Phi ^*_h\):

$$\begin{aligned} \partial _t \Phi ^*_h = \Delta _h \left( \left( \Phi ^*_h \right) ^3 + \left( \left[ J \star \mathbf{1} \right] + \gamma _c - \gamma _e \right) \Phi ^*_h - \left[ J \star \Phi ^*_h \right] \right) + O(h^4) , \quad \forall \ (i,j),\nonumber \\ \end{aligned}$$
(4.29)

for which the following estimate was used:

$$\begin{aligned} \left( \Phi ^*_h \right) ^3 = \left( \Phi + h^2 \Phi ^1_{h} \right) ^3 = \Phi ^3 + 3 h^2 \Phi ^2 \Phi ^1_h + O (h^4). \end{aligned}$$
(4.30)

We remark that the above derivation is valid since all \(O(h^2)\) terms cancel in the expansion.

Regarding the temporal correction term, we observe that the application of the convex splitting scheme (3.6) for the profile \(\Phi ^*_h\) gives

$$\begin{aligned} \frac{ \left( \Phi ^*_h \right) ^{k+1} - \left( \Phi ^*_h \right) ^k}{s}&= \Delta _h \Biggl ( \left( \left( \Phi ^*_h \right) ^{k+1} \right) ^3 + \left( 2 \left[ J_c \star \mathbf{1} \right] + \gamma _c \right) \left( \Phi ^*_h \right) ^{k+1} \nonumber \\&-\,\left( \left[ J_c \star \mathbf{1} \right] + \left[ J_e \star \mathbf{1} \right] + \gamma _e \right) \left( \Phi ^*_h \right) ^k -\left[ J \star \left( \Phi ^*_h \right) ^k \right] \Biggr )\nonumber \\&+\, s {{\varvec{h}}}^{(1)} + O (s^2) + O(h^4) , \quad \forall \ (i,j). \end{aligned}$$
(4.31)

In turn, the first order temporal correction function \(\Phi ^1_s\) is given by the solution of the following linear ordinary differential equations

$$\begin{aligned} \partial _t \Phi ^1_s&= \Delta _h \Bigl ( 3 \left( \Phi ^*_h \right) ^2 \Phi ^1_s + \left( \left[ J_c \star \mathbf{1} \right] - \left[ J_e \star \mathbf{1} \right] + \gamma _c - \gamma _e \right) \Phi ^1_s - \left[ J \star \Phi ^1_s \right] \Bigr ) - {{\varvec{h}}}^{(1)}.\nonumber \\ \end{aligned}$$
(4.32)

Again, the solution of (4.32), which exists and is unique, depends solely on the profile \(\Phi ^*_h\) and is smooth enough in the sense that its divided differences of various orders are bounded. Similar to (4.31), an application of the convex splitting scheme to \(\Phi ^1_s\) reads

$$\begin{aligned} \frac{ \left( \Phi ^1_s \right) ^{k+1} - \left( \Phi ^1_s \right) ^k}{s}&= \Delta _h \Biggl ( 3 \left( \left( \Phi _h^*\right) ^{k+1} \right) ^2 \left( \Phi _s^1\right) ^{k+1} + \left( 2 \left[ J_c \star \mathbf{1} \right] + \gamma _c \right) \left( \Phi ^1_s \right) ^{k+1} \nonumber \\&- \left( \left[ J_c \star \mathbf{1} \right] + \left[ J_e \star \mathbf{1} \right] + \gamma _e \right) \left( \Phi ^1_s \right) ^k - \left[ J \star \left( \Phi ^1_s \right) ^k \right] \Biggr )\nonumber \\&- ( {{\varvec{h}}}^{(1)} )^k + s ({{\varvec{h}}}^{(2)} )^k + O (s^2) + O(s h^2) , \quad \forall \ (i,j).\quad \quad \end{aligned}$$
(4.33)

Therefore, a combination of (4.31) and (4.33) shows that

$$\begin{aligned} \frac{ \hat{\Phi }^{k+1} \!-\! \hat{\Phi }^k}{s}&= \Delta _h \Bigl ( \left( \hat{\Phi }^{k+1} \right) ^3 \!+\! \left( 2 \left[ J_c \star 1 \right] \!+\! \gamma _c \right) \hat{\Phi }^{k+1} - \left( \left[ J_c \star 1 \right] \!+\! \left[ J_e \star 1 \right] + \gamma _e \right) \hat{\Phi }^k \nonumber \\&- \left[ J \star \hat{\Phi }^k \right] \Bigr ) + O (s^2 + s h^2 + h^4) , \quad \forall \ (i,j) , \end{aligned}$$
(4.34)

in which the construction (4.25) for the approximate solution \(\hat{\Phi }\) is recalled and we have used the following estimate

$$\begin{aligned} \hat{\Phi }^3 = \left( \Phi ^*_h + s \Phi ^1_s \right) ^3 = \left( \Phi ^*_h \right) ^3 + 3 s \left( \Phi ^*_h \right) ^2 \Phi _1^s + O (s^2). \end{aligned}$$
(4.35)

As stated earlier, the purpose of the higher order expansion (4.25) is to obtain an \(L^{\infty }\) estimate of the error function via its \(L^2\) norm in higher order accuracy by utilizing an inverse inequality in spatial discretization, which will be shown below. A detailed analysis shows that

$$\begin{aligned} \left\| \hat{\Phi } - \Phi \right\| _\infty \le C ( s + h^2 ) , \end{aligned}$$
(4.36)

since \(\left\| \Phi ^1_h \right\| _\infty , \left\| \Phi ^1_s \right\| _\infty \le C\). Subsequently, the following error function is considered:

$$\begin{aligned} \hat{e}^k_{i,j} := \hat{\Phi }^k_{i,j} - \phi ^k_{i,j}. \end{aligned}$$
(4.37)

In other words, instead of a direct comparison between the numerical solution \(\phi \) and the exact solution \(\Phi \), we estimate the error between the numerical solution and the constructed solution to obtain a higher order convergence in \(\left\| \, \cdot \, \right\| _2\) norm. Subtracting (3.6) from (4.35) yields

$$\begin{aligned} \hat{e}^{k+1} \!-\! \hat{e}^k = s\Delta _h\left( w\left( \hat{\Phi }^{k+1},\hat{\Phi }^k\right) \!-\! w\left( \phi ^{k+1},\phi ^k\right) \right) + s O (s^2 + s h^2 + h^4).\quad \quad \quad \end{aligned}$$
(4.38)

By repeating the same technique in the proof of Theorem 4.1, we arrive at the following error estimate in the \(\ell ^\infty (0,T; \ell ^2)\) norm:

$$\begin{aligned} \left\| {\hat{\Phi }}^\ell - \phi ^\ell \right\| _{2}= \left\| \hat{e}^l \right\| _{2} \le C \left( s^2 + s h^2 + h^4 \right) , \quad l \cdot s \le T. \end{aligned}$$
(4.39)

Therefore, by the inverse inequality (6.2), we get

$$\begin{aligned} \left\| \hat{\Phi }^\ell - \phi ^\ell \right\| _{\infty } \le \frac{C}{h}\left\| \hat{\Phi }^\ell - \phi ^\ell \right\| _2 \le \frac{ C \left( s^2 + s h^2 + h^4 \right) }{h} \le C \left( s + h^2 \right) ,\quad \quad \end{aligned}$$
(4.40)

since \(s \le C h\). The triangle inequality now yields

$$\begin{aligned} \left\| \Phi ^\ell - \phi ^\ell \right\| _\infty \le \left\| \hat{\Phi }^\ell - \Phi ^\ell \right\| _\infty + \left\| \hat{\Phi }^\ell - \phi ^\ell \right\| _\infty \le C ( s + h^2 ) , \end{aligned}$$
(4.41)

with an application of triangle inequality. The result is proven. \(\square \)

5 Numerical tests

In this section we discuss the results of two numerical experiments verifying the theoretical convergence and stability of the scheme. Numerical solutions of the scheme were calculated using a nonlinear multigrid solver, similar to the solvers described in [28, 40]. The details and performance of the solver will be given in a later paper. The first experiment verifies the convergence rate in the \(\ell ^\infty (0,T;\ell ^2)\) norm. We use smooth, periodic initial data \(\phi (x_1,x_2,0) = 0.5\sin (2\pi x_1)\sin (2 \pi x_2)\) on a square domain \(\Omega = (-0.5,0.5)^2\). The convolution kernel is constructed via

$$\begin{aligned} \mathsf {J}_c(x_1,x_2) = \alpha \exp \left( -\frac{x_1^2+x_2^2}{\sigma ^2}\right) , \quad \mathsf {J}_e\equiv 0 , \quad \sigma = 0.05, \quad \alpha = \frac{1}{\sigma ^2}. \end{aligned}$$
(5.1)

The composite kernel \(\mathsf {J}=\mathsf {J}_c-\mathsf {J}_e\) is very small in magnitude near the boundary of \(\Omega \), and we extend it periodically outside of \(\Omega \). Technically, \(\mathsf {J}\notin C_\mathrm{per}^\infty \), but this does not affect the convergence, as expected. The other parameters are \(\gamma _c =0\) and \(\gamma _e = 1\). The time step size is precisely \(s = h^2\), with the spatial step sizes as indicated in Table 1. The final time for the tests is taken as \(T=40\left( \frac{1}{128}\right) ^2 = 2.44140625\times 10^{-3}\).

Table 1 The difference between coarse and fine grids of the computed numerical solutions using a quadratic refinement path.
Full size table

As we utilize a quadratic refinement path, \(s = h^2\), the global error is expected to be \(O(h^2)\) in the \(\ell ^\infty (0,T;\ell ^2)\) norm. We do not have the exact solution—these are not easily obtained for non-trivial convolution kernels—against which to compare our computed solutions in the present case. To overcome this, in our error calculations we are using the difference between results on successive coarse and fine grids as a measure of the error. The difference function, \(e_{A}\), is evaluated at time \(t = T\) using the method described in [28, 40]. The result is given in Table 1. The global second-order accuracy of the method is confirmed when a quadratic refinement path is used.

The second experiment is the simulation of spinodal decomposition. The initial data are random, specifically, \(\phi (x_1,x_2,0) = r(x_1,x_2)\), where \(r\) is a uniformly distributed random variable in the interval \([-0.05,0.05]\). The convolution kernel \(\mathsf {J}\) is the same as in (5.1). We take \(\gamma _c = 0\), \(\gamma _e=1\) and \(\Omega = (-5,5)^2\). The parameters of the numerical simulation are \(h = 10/256\) and \(s = 1.0\times 10^{-4}\). The final time for the simulation is \(T=2\). Figure 1 shows snapshots of the evolution of \(\phi \) up to time \(T=2\), and Fig. 2 shows the corresponding numerical energy for the simulation. The energy is observed to decay as time increases, as predicted by the theory.

Fig. 1
figure 1

Spinodal decomposition of with random initial data using the nonlocal model. The plots above show filled contour plots of the computed solution \(\phi \) at the indicated times

Full size image
Fig. 2
figure 2

The computed numerical energy (3.1) as a function of time for the spinodal decomposition simulation depicted in Fig. 1

Full size image

Remark 5.1

To keep the current exposition brief, we have severely limited the number of computational examples in this paper. But, note that the assumptions on the convolution kernel, (A1)–(A3), are rather general, and our methods apply to a wide a variety of physically relevant kernels. Many more computational examples, including anisotropic, multi-well, and unsigned kernels, can be found in our recent paper [25].