Finite difference schemes for the parabolic p-Laplace equation

Abstract

We propose a new finite difference scheme for the degenerate parabolic equation

$$\begin{aligned} \partial _t u - \text{ div }(|\nabla u|^{p-2}\nabla u) =f, \quad p\ge 2. \end{aligned}$$

Under the assumption that the data is Hölder continuous, we establish the convergence of the explicit-in-time scheme for the Cauchy problem provided a suitable stability type CFL-condition. An important advantage of our approach, is that the CFL-condition makes use of the regularity provided by the scheme to reduce the computational cost. In particular, for Lipschitz data, the CFL-condition is of the same order as for the heat equation and independent of p.

1 Introduction

Recently, a new monotone finite difference discretization of the p-Laplacian was introduced by the authors in [9]. It is based on the mean value property presented in [4, 8]. The aim of this paper is to propose an explicit-in-time finite difference numerical scheme for the following Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u(x,t)-\Delta _pu(x,t) =f(x), &{} x\in {\mathbb {R}}^d\times {(0,T)},\\ u(x,0)=u_0(x), &{} x \in {\mathbb {R}}^d, \end{array}\right. } \end{aligned}$$

(1.1)

and study its convergence. Here, $p\ge 2$ and $\Delta _p$ is the p-Laplace operator,

$$\begin{aligned} \Delta _p\psi =\text{ div }(|\nabla \psi |^{p-2}\nabla \psi ). \end{aligned}$$

The main result is the pointwise convergence of our scheme given Hölder continuous data (f and $u_0$) and a stability type CFL-condition. See Theorem 2.2 for the precise statement and ($CFL$) for the CFL-condition. One of the advantages of our approach is that the CFL-condition makes use of the regularity provided by the scheme. As a consequence, for Lipschitz continuous data, the CFL-condition is of the same order as the one for the heat equation. In general, the order of the CFL-condition depends on p and on the regularity of the data.

1.1 Related literature

Equation (1.1) has attracted much attention in the last decades. We refer to [11, 13] for the theory for weak solutions of this equation and to [23] for the relation between viscosity solutions and weak solutions. To the best of our knowledge, the best regularity results known are $C^{1,\alpha }-$regularity in space for some $\alpha >0$ (see [11, Chapter IX]) and $C^{0,1/2}-$regularity in time (see [3, Theorem 2.3]).

The literature regarding finite difference schemes for parabolic problems involving the p-Laplacian is quite scarce. One reason for that is naturally that, since the p-Laplacian is in divergence form, it is very well suited for methods based on finite elements, see for instance [1, 2, 14, 19, 21] for related results.

In the stationary setting, there has been some development of finite difference methods the past 20 years. Section 1.1 in [28] provides an accurate overview of such results, we will only mention a few. In [5, 10, 18, 28], finite difference schemes for the p-Laplace equation based on the mean value formula for the normalized p-Laplacian (cf. [25]) are considered. Since the corresponding parabolic equation for the normalized p-Laplacian is completely different in nature (see [15, 22]), these methods do not seem very well suited to be used for the parabolic equation considered in this paper. In [9], the authors of the present paper studied a monotone finite difference discretization of the p-Laplacian based on the mean value property presented in [4, 8]. We also seize the opportunity to mention [27], where difference schemes for degenerate elliptic and parabolic equations (but not for equation (1.1)) are discussed.

It is noteworthy that, in dimension $d=1$, the spatial derivative of a solution of (1.1) is a solution of the Porous Medium Equation (PME). See [29, 30] for a general presentation of the PME, and [20] for a proof of this fact. Finite difference schemes for the PME are well known, see [6, 7, 12, 16, 26].

2 Assumptions and main results

In this section, we introduce a general form of finite difference discretizations of $\Delta _p$ and the associated numerical scheme for (1.1). This is followed by our assumptions, the notion of solutions for (1.1) and the formulation of our main result.

2.1 Discretization and scheme

In order to treat (1.1), we consider a general discretization of $\Delta _p$ of the form

$$\begin{aligned} D^h_p\psi (x)=\sum _{y_\beta \in {\mathcal {G}}_h} J_p(\psi (x+y_\beta )-\psi (x)) \omega _{\beta }, \end{aligned}$$

(2.1)

where

$$\begin{aligned} J_p(\xi )=|\xi |^{p-2}\xi ,\quad \xi \in {\mathbb {R}},\quad {\mathcal {G}}_h:=h{\mathbb {Z}}^d=\{y_\beta := h \beta \, : \, \beta \in {\mathbb {Z}}^d\} \end{aligned}$$

and $\omega _\beta$ are certain weights $\omega _\beta =\omega _\beta (h)$ satisfying $\omega _\beta =\omega _{-\beta }\ge 0$.

We also need to introduce a time discretization. We will employ an explicit and uniform-in-time discretization. Let $N\in {\mathbb {N}}$ and consider a discretization parameter $\tau >0$ given by $\tau =T/N$. Consider also the sequence of times $\{t_j\}_{j=0}^{N}$ defined by $t_0=0$ and $t_j=t_{j-1}+ \tau = j\tau$. The time grid, ${\mathcal {T}}_\tau$, is given by

$$\begin{aligned} {\mathcal {T}}_{\tau }= \bigcup _{j=0}^N \{t_j\}. \end{aligned}$$

Then, our general form of an explicit finite difference scheme of (1.1) is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} U^j_\alpha = U_\alpha ^{j-1} +\tau \left( D_p^h U_{\alpha }^{j-1}+f_\alpha \right) , &{} \alpha \in {\mathbb {Z}}^d,\, j=1,\ldots ,N,\\ U^0_\alpha =(u_0)_\alpha &{} \alpha \in {\mathbb {Z}}^d, \end{array}\right. } \end{aligned}$$

(2.2)

where $f_\alpha :=f(x_\alpha )$, $(u_0)_\alpha =u_0(x_\alpha )$ and $D_p^h$ is given by (2.1).

2.2 Assumptions

In order to ensure convergence of the scheme (2.2), we impose the following hypotheses on the data and the discretization parameters. This entails a regularity assumption on the data, some assumptions on the discretization and a nonlinear CFL-condition on the parameters, as is customary for explicit schemes.

Hypothesis on the data. We assume that

More precisely,

$$\begin{aligned} |u_0(x)-u_0(y)|\le L_{u_0}|x-y|^a \quad and \quad |f(x)-f(y)|\le L_{f}|x-y|^a, \qquad for all \quad x,y\in {\mathbb {R}}^d, \end{aligned}$$

for some constants $L_{u_0},L_{f}\ge 0$. Sometimes we will write $\Lambda _{u_0}(\delta ):=L_{u_0}\delta ^a$ and $\Lambda _{f}(\delta ):=L_{f}\delta ^a$ to simplify the presentation.

Hypothesis on the spatial discretization. For the discretization, we assume the following type of monotonicity and boundedness:

Here $M=M(p,d)>0$. In addition, we assume the following consistency for the discretization:

Examples of discretizations satisfying these properties can be found in Sect. 5.

Hypothesis on the discretization parameters. We assume the following stability condition on the numerical parameters:

with

$$\begin{aligned} C=\min \left\{ 1, \frac{1}{M (p-1)\left( L_{u_0}+TL_f +3{\tilde{K}}+1\right) ^{p-2}}\right\} \end{aligned}$$

and ${\tilde{K}}$ a constant given in (3.4), depending on p, the modulus of continuity in time of the discretized solution and some universal constants coming from a mollifier.

Remark 2.1

For Lipschitz data $u_0$ and f, the condition ($CFL$) reads $\tau \le {Cr^2}$ for a certain constant $C=C(u_0,f,d,p,T)>0$. We note that, regardless of the constant C, the relation between $\tau$ and r is always quadratic (as in the linear case $p=2$) and independent of p. It is important to mention that this is computationally very relevant, especially if we want to deal with problems related to large p.

2.3 Main result

We now state our main result regarding the convergence of the scheme. Several other properties of the scheme are also obtained, but we will state them later.

Theorem 2.2

Let $p\in [2,\infty )$ and assume ($A _{u_0,f}$) and ($A _\omega$). Then for every $h,\tau >0$, there exists a unique solution $U\in \ell ^\infty ({\mathcal {G}}_h\times {\mathcal {T}}_\tau )$ of (2.2). If the hypotheses ($CFL$) and ($A _{c}$) additionally hold, then

$$\begin{aligned} \max _{(x_\alpha ,t_j)\in {\mathcal {G}}_h\times {\mathcal {T}}_\tau }|U_\alpha ^j- u(x_\alpha ,t_j)|\rightarrow 0 \quad as \quad h\rightarrow 0^+, \end{aligned}$$

where u is the unique viscosity solution of (1.1).

2.4 Viscosity solutions

Throughout the paper, we will use the notion of viscosity solutions. For completeness, we define the concept of viscosity solutions of (1.1), adopting the definition in [23].

Definition 2.1

Assume ($A _{u_0,f}$). We say that a bounded lower (resp. upper) semicontinuous function u in ${\mathbb {R}}^d\times [0,T]$ is a viscosity supersolution (resp. subsolution) of (1.1) if

• (a) $u(x,0)\ge u_0(x)$ (resp. $u(x,0)\le u_0(x)$);

• (b) whenever $(x_0,t_0)\in {\mathbb {R}}^d\times (0,T)$ and $\varphi \in C^2_b(B_R(x_0)\times (t_0-R,t_0+R))$ for some $R>0$ are such that $\varphi (x_0,t_0)=u(x_0,t_0)$ and $\varphi (x,t)< u(x,t)$ (resp. $\varphi (x,t)> u(x,t)$) for $(x,t) \in B_R(x_0)\times (t_0-R,t_0)$, then we have

  $$\begin{aligned} \varphi _t(x_0,t_0)-\Delta _p\varphi (x_0,t_0)\ge f(x_0) \quad (\text {resp.} \quad \varphi _t(x_0,t_0)-\Delta _p\varphi (x_0,t_0)\le f(x_0) ). \end{aligned}$$

A viscosity solution of (1.1) is a bounded continuous function u being both a viscosity supersolution and a viscosity subsolution (1.1).

Remark 2.3

We remark that it is not necessary to require strict inequality in the definition above. It is enough to require $\varphi (x,t)\le u(x,t)$ (resp. $\varphi (x,t)\ge u(x,t)$) for $(x,t) \in B_R(x_0)\times (t_0-R,t_0)$.

We also state a necessary uniqueness result that will ensure convergence of the scheme. Without such a result, we would only be able to establish convergence up to a subsequence. The theorem below is a consequence of the fact that viscosity solutions are weak solutions (see Corollary 4.7 in [23]) and that bounded weak solutions are unique (see Theorem 6.1 in [11]).

Theorem 2.4

Assume ($A _{u_0,f}$). Then there is a unique solution of (1.1).

3 Properties of the numerical scheme

In this section we will study properties of the numerical scheme (2.2). More precisely, we establish existence and uniqueness for the numerical solution, stability in maximum norm, as well as conservation of the modulus of continuity of the data.

3.1 Existence and uniqueness

We have the following existence and uniqueness result for the numerical scheme.

Proposition 3.1

Assume ($A _{u_0,f}$), ($A _\omega$), $p\ge 2$ and $r,h,\tau >0$. Then there exists a unique solution $U\in \ell ^\infty ({\mathcal {G}}_h\times {\mathcal {T}}_{\tau })$ of the scheme (2.2).

Proof

First we note that, for a function $\psi \in \ell ^\infty ({\mathcal {G}}_h)$, we have that

$$\begin{aligned} | D^h_p\psi _\alpha |\le \sum _{y_\beta \in {\mathcal {G}}_h} J_p(\psi (x_\alpha +y_\beta )-\psi (x_\alpha )) \omega _{\beta } \le (2\Vert \psi \Vert _{\ell ^\infty ({\mathcal {G}}_h)})^{p-1}\sum _{y_\beta \in {\mathcal {G}}_h}\omega _{\beta }<+\infty . \end{aligned}$$

Then, for each $\alpha \in {\mathbb {Z}}^d$, $U^j_\alpha$ is defined recursively using the values of $U^{j-1}_\beta$ for $\beta \in {\mathbb {Z}}^d$, and we have that

$$\begin{aligned} \sup _{y_\alpha \in {\mathcal {G}}_h}|U^j_\alpha |= \sup _{y_\alpha \in {\mathcal {G}}_h}|U^{j-1}_\alpha | + \tau \left( \left( 2\sup _{y_\alpha \in {\mathcal {G}}_h}|U^{j-1}_\alpha |\right) ^{p-1} \sum _{y_\beta \in {\mathcal {G}}_h}\omega _{\beta } + \sup _{y_\alpha \in {\mathcal {G}}_h}|f_\alpha |\right) . \end{aligned}$$

The conclusion follows since

$$\begin{aligned} \sup _{y_\alpha \in {\mathcal {G}}_h}|f_\alpha |\le \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)} \quad and \quad \sup _{y_\alpha \in {\mathcal {G}}_h}|U^0_\alpha |=\sup _{y_\alpha \in {\mathcal {G}}_h}|u_0(y_\alpha )|\le \Vert u_0\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned}$$

$\square$

3.2 Stability and preservation of the modulus of continuity in space

First we will prove that the scheme preserves the regularity of the data.

Proposition 3.2

Assume ($A _{u_0,f}$), ($A _\omega$), $p\ge 2$, $r, h, \tau >0$ and ($CFL$). Let U be the solution of (2.2). For every $j=0,\ldots ,N$, we have

$$\begin{aligned} |U^j_\alpha -U^j_\gamma | \le \Lambda _{u_0}(|x_\alpha -x_\gamma |) + t_j \Lambda _f(|x_\alpha -x_\gamma |), \quad for all \quad x_\alpha ,x_\gamma \in {\mathcal {G}}_h. \end{aligned}$$

Proof

By assumption ($A _{u_0,f}$), for any given $x_\alpha ,x_\gamma \in {\mathcal {G}}_h$, we have that

$$\begin{aligned} |U^0_\alpha -U^0_\gamma |= |u_0(x_\alpha )-u_0(x_\gamma )|\le \Lambda _{u_0}(|x_\alpha -x_\gamma |). \end{aligned}$$

Assume by induction that

$$\begin{aligned} |U^{j}_\alpha -U^{j}_\gamma |\le \Lambda _{u_0}(|x_\alpha -x_\gamma |) + t_j \Lambda _f(|x_\alpha -x_\gamma |). \end{aligned}$$

Using the scheme at $x_\alpha$ and $x_\gamma$ we get

$$\begin{aligned} U^{j+1}_\alpha -U^{j+1}_\gamma = U^{j}_\alpha -U^{j}_\gamma +\tau \sum _{y_\beta \in {\mathcal {G}}_h} \left( J_p( U^{j}_{\alpha +\beta }-U^{j}_{\alpha }) - J_p( U^{j}_{\gamma +\beta }-U^{j}_{\gamma })\right) \omega _\beta + \tau (f_\alpha -f_\gamma ). \end{aligned}$$

Now, since $p\ge 2$, we have, by Taylor expansion, that

$$\begin{aligned} J_p( U^{j}_{\alpha +\beta }-U^{j}_{\alpha }) - J_p( U^{j}_{\gamma +\beta }-U^{j}_{\gamma })=(p-1)|\eta _\beta |^{p-2} \left( (U^{j}_{\alpha +\beta }-U^{j}_{\gamma +\beta })-(U^{j}_{\alpha }-U^{j}_{\gamma })\right) , \end{aligned}$$

for some $\eta _\beta \in {\mathbb {R}}$ between $(U^{j}_{\alpha +\beta }-U^{j}_{\alpha })$ and $(U^{j}_{\gamma +\beta }-U^{j}_{\gamma })$. Thus,

$$\begin{aligned} \begin{aligned} U^{j+1}_\alpha -U^{j+1}_\gamma =&(U^{j}_\alpha -U^{j}_\gamma )\left( 1-\tau (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} \omega _\beta \right) \\&+\tau (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} (U^{j}_{\alpha +\beta }-U^{j}_{\gamma +\beta }) \omega _\beta + \tau (f_\alpha -f_\gamma ). \end{aligned} \end{aligned}$$

(3.1)

Now observe that, by the induction assumption, we have

$$\begin{aligned} |\eta _\beta |{} & {} \le \sup _{y_\alpha \in {\mathcal {G}}_h} \{|U^{j}_{\alpha +\beta }-U^{j}_{\alpha }|\}\le \sup _{y_\alpha \in {\mathcal {G}}_h} \{\Lambda _{u_0}(|x_{\alpha +\beta }-x_\alpha |) + t_j \Lambda _f(|x_{\alpha +\beta }-x_\alpha |)\}\\{} & {} = \Lambda _{u_0}(| x_{\beta }|) + t_j \Lambda _f( |x_{\beta }|). \end{aligned}$$

By ($A _\omega$), we have $w_\beta =0$ for $y_\beta \not \in B_r$ for some $r>0$, and we deduce that

$$\begin{aligned} \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} \omega _\beta \le \left( \Lambda _{u_0}(r) + t_j \Lambda _f(r)\right) ^{p-2}\sum _{y_\beta \in {\mathcal {G}}_h}\omega _{\beta }\le \frac{(L_{u_0}+t_j L_f)^{p-2}M}{ r^{2+(1-a)(p-2)}}. \end{aligned}$$

Thus, by ($CFL$), we get

$$\begin{aligned} \tau (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} \omega _\beta \le 1. \end{aligned}$$

Using the above estimate and the induction hypothesis in (3.1), we get that

$$\begin{aligned} \begin{aligned}\ |U^{j+1}_\alpha -U^{j+1}_\gamma |\le&|U^{j}_\alpha -U^{j}_\gamma |\left( 1-\tau (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} \omega _\beta \right) \\&+\tau (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} |U^{j}_{\alpha +\beta }-U^{j}_{\gamma +\beta }| \omega _\beta + \tau |f_\alpha -f_\gamma |\\ \le&\left( \Lambda _{u_0}(|x_\alpha -x_\gamma |) + t_j \Lambda _f(|x_\alpha -x_\gamma |)\right) \left( 1-\tau (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} \omega _\beta \right) \\&+\tau (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} \Big (\Lambda _{u_0}(|x_{\alpha +\beta }-x_{\gamma +\beta }|) \\&+ t_j \Lambda _f(|x_{\alpha +\beta }-x_{\gamma +\beta }|)\Big ) \omega _\beta + \tau \Lambda _f (|x_\alpha -x_\gamma |)\\ \le&\left( \Lambda _{u_0}(|x_\alpha -x_\gamma |) + t_j \Lambda _f(|x_\alpha -x_\gamma |)\right) \left( 1-\tau (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} \omega _\beta \right) \\&+\tau (p-1) \left( \Lambda _{u_0}(|x_\alpha -x_\gamma |) + t_j \Lambda _f(|x_\alpha -x_\gamma |)\right) \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} \omega _\beta \\&+ \tau \Lambda _f (|x_\alpha -x_\gamma |)\\ =&\Lambda _{u_0}(|x_\alpha -x_\gamma |) + (t_j+\tau ) \Lambda _f(|x_\alpha -x_\gamma |), \end{aligned} \end{aligned}$$

which concludes the proof. $\square$

Remark 3.3

In particular, if both $u_0$ and f are Lipschitz functions with constants $L_{u_0}$ and $L_f$ respectively, the above result reads,

$$\begin{aligned} |U^j_\alpha -U^j_\gamma | \le (L_{u_0}+t_j L_f)|x_\alpha -x_\gamma |. \end{aligned}$$

We are now ready to state and prove the stability result: solutions with bounded data remain bounded (uniformly in the discretization parameters) for all times.

Proposition 3.4

Under the assumptions of Proposition 3.2, we have that

$$\begin{aligned} \sup _{y_\alpha \in {\mathcal {G}}_h} |U^j_\alpha |\le \Vert u_0\Vert _{L^\infty ({\mathbb {R}}^d)} + t_j \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}, \quad for all \quad j=0,\ldots , N. \end{aligned}$$

Proof

By assumption ($A _{u_0,f}$), we have that

$$\begin{aligned} \sup _{y_\alpha \in {\mathcal {G}}_h}|U^0_\alpha |\le \sup _{y_\alpha \in {\mathcal {G}}_h}|u_0(x_\alpha )| \le \Vert u_0\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned}$$

Assume by induction that

$$\begin{aligned} \sup _{y_\alpha \in {\mathcal {G}}_h} |U^j_\alpha |\le \Vert u_0\Vert _{L^\infty ({\mathbb {R}}^d)} + t_j \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned}$$

Direct computations lead to

$$\begin{aligned} \begin{aligned} U^{j+1}_\alpha&=U^{j}_\alpha + \tau \sum _{y_\beta \in {\mathcal {G}}_h} |U^{j}_{\alpha +\beta }-U^{j}_\alpha |^{p-2}(U^{j}_{\alpha +\beta }-U^{j}_\alpha )\omega _\beta + \tau f_\alpha \\&=U^{j}_\alpha \left( 1-\tau \sum _{y_\beta \in {\mathcal {G}}_h} |U^{j}_{\alpha +\beta }-U^{j}_\alpha |^{p-2} \omega _\beta \right) + \tau \sum _{y_\beta \in {\mathcal {G}}_h} |U^{j}_{\alpha +\beta }-U^{j}_\alpha |^{p-2}U^{j}_{\alpha +\beta }\omega _\beta + \tau f_\alpha . \end{aligned} \end{aligned}$$

By Proposition 3.2 we have that

$$\begin{aligned} |U^{j}_{\alpha +\beta }-U^{j}_\alpha |^{p-2}\le (\Lambda _{u_0}(|y_\beta |) + t_j \Lambda _f(|y_\beta |))^{p-2}, \end{aligned}$$

which together with assumptions ($A _\omega$) and ($CFL$) imply that

$$\begin{aligned} \tau \sum _{y_\beta \in {\mathcal {G}}_h} |U^{j}_{\alpha +\beta }-U^{j}_\alpha |^{p-2} \omega _\beta \le \tau (\Lambda _{u_0}(r) + t_j \Lambda _f(r))^{p-2}\sum _{y_\beta \in {\mathcal {G}}_h} \omega _\beta \le \frac{1}{p-1}\le 1. \end{aligned}$$

Direct computations plus the induction hypothesis allow us to conclude that

$$\begin{aligned} \begin{aligned} |U^{j+1}_\alpha |\le&\sup _{y_\alpha \in {\mathcal {G}}_h} |U^{j}_\alpha |\left( 1-\tau \sum _{y_\beta \in {\mathcal {G}}_h} |U^{j}_{\alpha +\beta }-U^{j}_\alpha |^{p-2} \omega _\beta \right) \\&+ \tau \sup _{y_\alpha \in {\mathcal {G}}_h} |U^{j}_\alpha | \sum _{y_\beta \in {\mathcal {G}}_h} |U^{j}_{\alpha +\beta }-U^{j}_\alpha |^{p-2}\omega _\beta + \tau \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}\\ =&\sup _{y_\alpha \in {\mathcal {G}}_h} |U^{j}_\alpha | + \tau \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}\\ =&\Vert u_0\Vert _{L^\infty ({\mathbb {R}}^d)} + (t_j +\tau ) \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}, \end{aligned} \end{aligned}$$

which concludes the proof. $\square$

3.3 Time equicontinuity for a discrete in time scheme

Now we extend the scheme from ${\mathcal {G}}_h$ to ${\mathbb {R}}^d$ by considering $U:{\mathbb {R}}^d\times {\mathcal {T}}_{\tau }$ defined by

$$\begin{aligned} {\left\{ \begin{array}{ll} U^j(x)= U^{j-1}(x) + \tau \left( D_p^h U^{j-1}(x)+f(x)\right) , &{} x \in {\mathbb {R}}^d,\, j=1,\ldots ,N,\\ U^0(x)=u_0(x) &{} x \in {\mathbb {R}}^d. \end{array}\right. } \end{aligned}$$

(3.2)

Remark 3.5

Clearly, if we restrict the solution of (3.2) to ${\mathcal {G}}_h$, we recover the solution of (2.2).

Proposition 3.6

(Continuous dependence on the data) Assume ($A _{u_0,f}$), ($A _\omega$), $p\ge 2$, $r, h,\tau >0$ and ($CFL$). Let $U,{\widetilde{U}}$ be the solutions of (2.2) corresponding to $u_0, {\widetilde{u}}_0$ and $f,{\widetilde{f}}$. For every $j=0,\ldots ,N$, we have

$$\begin{aligned} \Vert U^j-{\widetilde{U}}^j\Vert _{L^\infty ({\mathbb {R}}^d)} \le \Vert u_0- {\widetilde{u}}_0\Vert _{L^\infty ({\mathbb {R}}^d)} + t_j \Vert f- {\widetilde{f}}\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned}$$

Proof

By assumption ($A _{u_0,f}$), we have that

$$\begin{aligned} \Vert U^0-{\widetilde{U}}^0\Vert _{L^\infty ({\mathbb {R}}^d)}= \Vert u_0-{\widetilde{u}}_0\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned}$$

Assume by induction that

$$\begin{aligned} \Vert U^j-{\widetilde{U}}^j\Vert _{L^\infty ({\mathbb {R}}^d)}= \Vert u_0-{\widetilde{u}}_0\Vert _{L^\infty ({\mathbb {R}}^d)} + t_j \Vert f-{\widetilde{f}}\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned}$$

Similar computations as the ones in the proof of Proposition 3.2 yield

$$\begin{aligned} \begin{aligned} U^{j+1}(x)-{\widetilde{U}}^{j+1}(x)&= (U^{j}(x)-{\widetilde{U}}^{j}(x))\left( 1-\tau (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} \omega _\beta \right) +\tau (p-1) \\&\times \sum _{y_\beta \in {\mathcal {G}}_h} |\eta _\beta |^{p-2} (U^{j}(x+y_\beta )-{\widetilde{U}}^{j}(x+y_\beta )) \omega _\beta + \tau (f(x)-{\widetilde{f}}(x)), \end{aligned} \end{aligned}$$

(3.3)

where $\eta _\beta \in {\mathbb {R}}$ is some number between $(U^{j}(x+y_\beta )-U^{j}(x))$ and $({\widetilde{U}}^{j}(x+y_\beta )-{\widetilde{U}}^{j}(x))$. From here, the proof follows as in the proof of Proposition 3.2. $\square$

Proposition 3.7

(Equicontinuity in time) Assume ($A _{u_0,f}$), ($A _\omega$), $p\ge 2$, $r,h,\tau >0$ and ($CFL$). Let U be the solution of (3.2). Then

$$\begin{aligned} \begin{aligned} \Vert U^{j+k}-U^{j}\Vert _{L^\infty ({\mathbb {R}}^d)}&\le {\widetilde{K}} (t_k)^{\frac{a}{2+(1-a)(p-2)}} + \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}t_k=:{\overline{\Lambda }}_{u_0,f}(t_k), \end{aligned} \end{aligned}$$

with

$$\begin{aligned} {\tilde{K}} = 4^{\frac{1+(1-a)(p-1)}{2+(1-a)(p-2)}} L_{u_0}^{\frac{p}{2+(1-a)(p-2)}}((p-1) K_1^{p-2}K_2 M)^\frac{a}{2+(1-a)(p-2)}, \end{aligned}$$

(3.4)

where M comes from assumption ($A _\omega$), and $K_1$ and $K_2$ are constants given in Sect. 1 (depending on a certain choice of mollifiers).

Proof

Consider a mollification of the initial data $u_{0,\delta }=u_0* \rho _\delta$ where $\rho _\delta (x)$ is a standard mollifier (as defined in Appendix A). Let $(U_\delta )^j$ be the corresponding solution of (3.2) with $u_{0,\delta }$ as initial data. Then,

$$\begin{aligned} \Vert (U_\delta )^1-(U_\delta )^0\Vert _{L^\infty ({\mathbb {R}}^d)} \le \tau \Vert D_p^hu_{0,\delta }\Vert _{L^\infty ({\mathbb {R}}^d)} + \tau \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned}$$

Define ${\widetilde{U}}^j_\delta := U^{j+1}_\delta$ for all $j=0,\ldots ,N$. Clearly, ${\widetilde{U}}^j_\delta$ is the unique solution of (3.2) with initial data ${\widetilde{U}}^0_\delta =U^{1}_\delta$ and right hand side f. By Proposition 3.6

$$\begin{aligned} \begin{aligned} \Vert U^{j+1}_\delta -U^j_\delta \Vert _{L^\infty ({\mathbb {R}}^d)}&=\Vert {\widetilde{U}}^{j}_\delta -U^j_\delta \Vert _{L^\infty ({\mathbb {R}}^d)} \le \Vert {\widetilde{U}}^{0}_\delta -U^0_\delta \Vert _{L^\infty ({\mathbb {R}}^d)}=\Vert U^1_\delta -U^0_\delta \Vert _{L^\infty ({\mathbb {R}}^d)} \\&\le \tau \Vert D_p^hu_{0,\delta }\Vert _{L^\infty ({\mathbb {R}}^d)} + \tau \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned} \end{aligned}$$

A repeated use of the triangle inequality yields

$$\begin{aligned} \begin{aligned} \Vert U^{j+k}_\delta -U^j_\delta \Vert _{L^\infty ({\mathbb {R}}^d)}&\le \sum _{i=0}^{k-1} \Vert U^{j+i+1}_\delta -U^{j+i}_\delta \Vert _{L^\infty ({\mathbb {R}}^d)}\\&\le (k\tau ) \Vert D_p^hu_{0,\delta }\Vert _{L^\infty ({\mathbb {R}}^d)} + (k\tau ) \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned} \end{aligned}$$

(3.5)

The symmetry of the weights $\omega _\beta$ together with Lemma B.1 implies

$$\begin{aligned} \begin{aligned} |D_p^hu_{0,\delta }(x)|&=\frac{1}{2} \left| \sum _{y_\beta \in {\mathcal {G}}_h} \left( J_p(u_{0,\delta }(x+y_\beta )-u_{0,\delta }(x))-J_p(u_{0,\delta }(x)-u_{0,\delta }(x-y_\beta ))\right) \omega _{\beta }\right| \\&\le \frac{p-1}{2} \sum _{y_\beta \in {\mathcal {G}}_h} \max \{|u_{0,\delta }(x+y_\beta )-u_{0,\delta }(x)|,|u_{0,\delta }(x)-u_{0,\delta }(x-y_\beta )|\}^{p-2} \\&\quad \times \left| u_{0,\delta }(x+y_\beta )+u_{0,\delta }(x-y_\beta )-2u_{0,\delta }(x) \right| \omega _\beta . \end{aligned} \end{aligned}$$

(3.6)

Now note that, by the a-Hölder regularity of $u_0$ given by assumption ($A _{u_0,f}$), Lemma A.1 and Lemma A.2 imply

$$\begin{aligned}{} & {} |u_{0,\delta }(x\pm y_\beta )-u_{0,\delta }(x)|\le K_1L_{u_0}\delta ^{a-1}|y_\beta |, \quad |u_{0,\delta }(x+y_\beta )+u_{0,\delta }(x-y_\beta )-2u_{0,\delta }(x) |\nonumber \\{} & {} \quad \le K_2 L_{u_0} \delta ^{a-2}|y_\beta |^2, \end{aligned}$$

(3.7)

where $K_1$ and $K_2$ depend only on the mollifier $\rho$. Now note that, by ($A _\omega$), we have

$$\begin{aligned} \sum _{y_\beta \in {\mathcal {G}}_h} |y_\beta |^p\omega _\beta \le M. \end{aligned}$$

(3.8)

Combining (3.5) and (3.8), we obtain

$$\begin{aligned} \begin{aligned} \Vert U^{j+k}_\delta -U^j_\delta \Vert _{L^\infty ({\mathbb {R}}^d)}&\le \frac{p-1}{2} t_k (K_1L_{u_0}\delta ^{a-1})^{p-2} K_2 L_{u_0} \delta ^{a-2}\sum _{y_\beta \in {\mathcal {G}}_h} |y_\beta |^p\omega _\beta + t_k \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}\\&\le {\widehat{K}} \delta ^{(a-1)(p-2)+(a-2)}t_k+ \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}t_k, \end{aligned} \end{aligned}$$

with ${\widehat{K}}=\frac{p-1}{2} K_1^{p-2}K_2 L_{u_0}^{p-1}M$. Using the triangle inequality, the above estimate and applying Proposition 3.6 several times we obtain

$$\begin{aligned} \begin{aligned} \Vert U^{j+k}-U^{j}\Vert _{L^\infty ({\mathbb {R}}^d)}&\le \Vert U^{j+k}-U^{j+k}_\delta \Vert _{L^\infty ({\mathbb {R}}^d)}+\Vert U^{j+k}_\delta -U^{j}_\delta \Vert _{L^\infty ({\mathbb {R}}^d)}+\Vert U^{j}-U^{j}_\delta \Vert _{L^\infty ({\mathbb {R}}^d)}\\&\le 2\Vert u_0-u_{0,\delta }\Vert _{L^\infty ({\mathbb {R}}^d)}+{\widehat{K}} \delta ^{(a-1)(p-2)+(a-2)}t_k+ \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}t_k\\&\le 2 L_{u_0}\delta ^a +{\widehat{K}} \delta ^{(a-1)(p-2)+(a-2)}t_k+ \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}t_k. \end{aligned} \end{aligned}$$

By choosing $\delta =(\frac{{\widehat{K}}}{2L_{u_0}}t_k)^{\frac{1}{2+(1-a)(p-2)}}$ in the above estimate, we get the desired result

$$\begin{aligned} \Vert U^{j+k}-U^{j}\Vert _{L^\infty ({\mathbb {R}}^d)} \le {\tilde{K}} (t_k)^{\frac{a}{2+(1-a)(p-2)}} + \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}t_k, \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} {\tilde{K}}&=4L_{u_0} \left( \frac{{\widehat{K}}}{2L_{u_0}} \right) ^\frac{a}{2+(1-a)(p-2)}\\&=4^{1-\frac{a}{2+(1-a)(p-2)}}L_{u_0}((p-1) K_1^{p-2}K_2 L_{u_0}^{p-2}M)^\frac{a}{2+(1-a)(p-2)}\\&= 4^{\frac{2+(1-a)(p-2) -a}{2+(1-a)(p-2)}} L_{u_0}^{\frac{p}{2+(1-a)(p-2)}}((p-1) K_1^{p-2}K_2 M)^\frac{a}{2+(1-a)(p-2)}. \end{aligned} \end{aligned}$$

$\square$

Remark 3.8

Actually, a close inspection of the previous proof reveals that for $u_0\in C^2_b({\mathbb {R}}^d)$ we can get

$$\begin{aligned} \Vert U^{j+k}-U^{j}\Vert _{L^\infty ({\mathbb {R}}^d)} \lesssim t_k. \end{aligned}$$

3.4 Equiboundedness and equicontinuity estimates for a scheme in ${\mathbb {R}}^d\times [0,T]$

We now need to extend the numerical scheme in time in a continuous way. This is done by continuous interpolation, i.e.,

$$\begin{aligned} U(x,t):= \frac{t_{j+1}-t}{\tau } U^j(x)+ \frac{t-t_j}{\tau }U^{j+1}(x) \quad if \quad t\in [t_j,t_{j+1}] \quad for some \quad j=0,\ldots ,N, \end{aligned}$$

(3.9)

where $U^j$ is the solution of (3.2).

Remark 3.9

It is standard to check that, for all $t\in [t_j,t_{j+1}]$, we have that the original scheme is preserved also outside the grid points, i.e.,

$$\begin{aligned} U(x,t)=U(x,t_j)+(t-t_j) D_p^h U(x,t_j)+ (t-t_j) f(x). \end{aligned}$$

(3.10)

We have the following result.

Proposition 3.10

(Stability and equicontinuity) Assume ($A _{u_0,f}$), ($A _\omega$), $p\ge 2$, $r,h,\tau >0$ and ($CFL$). Let U be the solution of (3.9). Then, we have:

• (a) (Equiboundedness) $\Vert U\Vert _{L^\infty ({\mathbb {R}}^N\times [0,T])} \le \Vert u_0\Vert _{L^\infty ({\mathbb {R}}^d)} +T \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}$,

• (b) (Equicontinuity) For any $x,z\in {\mathbb {R}}^d$ and $t,{\tilde{t}}\in [0,T]$ we have that

  $$\begin{aligned} |U(x,t)-U(z,{\tilde{t}})| \le \Lambda _{u_0}(|x-z|) +T \Lambda _{f}(|x-z|) + 3 {\overline{\Lambda }}_{u_0,f}(|{\tilde{t}}-t|). \end{aligned}$$

Proof

Equiboundedness follows easily from a continuous in space version of Proposition 3.4, since

$$\begin{aligned} \begin{aligned} |U(x,t)|&\le \frac{t_{j+1}-t}{\tau } \sup _{x\in {\mathbb {R}}^d} |U^j(x)|+ \frac{t-t_j}{\tau } \sup _{x\in {\mathbb {R}}^d}|U^{j+1}(x)|\\&\le \frac{t_{j+1}-t}{\tau } \left( \Vert u_0\Vert _{L^\infty ({\mathbb {R}}^d)} + T \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}\right) + \frac{t-t_j}{\tau } \left( \Vert u_0\Vert _{L^\infty ({\mathbb {R}}^d)} + T \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}\right) \\&\le \Vert u_0\Vert _{L^\infty ({\mathbb {R}}^d)} +T \Vert f\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned} \end{aligned}$$

Equicontinuity in space follows from the translation invariance of the scheme and Proposition 3.6:

$$\begin{aligned} |U(x+y,t)-U(x,t)|\le \Vert u_0(\cdot +y)-u_0\Vert _{L^\infty ({\mathbb {R}}^d)} +T \Vert f(\cdot +y)-f\Vert _{L^\infty ({\mathbb {R}}^d)}. \end{aligned}$$

To prove equicontinuity in time, we first consider $t,{\tilde{t}} \in [t_j,t_{j+1}]$ for some $j=0,\ldots ,N-1$. In this case we have

$$\begin{aligned} \begin{aligned} U(x,t)-U(x,{\tilde{t}})&=\left( \frac{t_{j+1}-t}{\tau } U^j(x)+ \frac{t-t_j}{\tau }U^{j+1}(x)\right) \\&\qquad -\left( \frac{t_{j+1}-{\tilde{t}}}{\tau } U^j(x)+ \frac{{\tilde{t}}-t_j}{\tau }U^{j+1}(x)\right) \\&= \frac{t-{\tilde{t}}}{\tau } \left( U^{j+1}(x)- U^j(x)\right) . \end{aligned} \end{aligned}$$

Then, from Proposition 3.7, we get

$$\begin{aligned} |U(x,t)-U(x,{\tilde{t}})|\le |t-{\tilde{t}}| \frac{ {\overline{\Lambda }}_{u_0,f}(\tau ) }{\tau } \end{aligned}$$

Note that the function $g(\tau )=\frac{ {\overline{\Lambda }}_{u_0,f}(\tau )}{\tau }$ is decreasing. Thus, since $|t-{\tilde{t}}|\le \tau$, we have $g(\tau )\le g(|t-{\tilde{t}}|)$. It follows that

$$\begin{aligned} \begin{aligned} |U(x,t)-U(x,{\tilde{t}})|&\le {\overline{\Lambda }}_{u_0,f}(|t-{\tilde{t}}|). \end{aligned} \end{aligned}$$

Now consider $t\in [t_j,t_{j+1})$ and ${\tilde{t}}\in [t_{j+k},t_{j+k+1})$ for $k\ge 1$. By the triangle inequality, the previous step and Proposition 3.7

$$\begin{aligned} \begin{aligned}&|U(x,t)-U(x,{\tilde{t}})|\\&\qquad \le |U(x,t)-U(x,t_{j+1})|+|U(x,t_{j+k})-U(x,{\tilde{t}})|+|U(x,t_{j+1})-U(x,t_{j+k})|\\&\qquad \le {\overline{\Lambda }}_{u_0,f}(|t_{j+1}-t|) + {\overline{\Lambda }}_{u_0,f}(|{\tilde{t}}-t_{j+k}|) +{\overline{\Lambda }}_{u_0,f}(|t_{j+k}-t_{j+1}|) . \end{aligned} \end{aligned}$$

Since $t\le t_{j+1} \le {\tilde{t}}$ and $t\le t_{j+k} \le {\tilde{t}}$, the above estimate yields

$$\begin{aligned} |U(x,t)-U(x,{\tilde{t}})|\le 3 {\overline{\Lambda }}_{u_0,f}(|{\tilde{t}}-t|) . \end{aligned}$$

Finally, we conclude space-time equicontinuity combining the above estimates to get

$$\begin{aligned} \begin{aligned} |U(x,t)-U(z,{\tilde{t}})|&\le |U(x,t)-U(z,t)|+|U(z,t)-U(z,{\tilde{t}})|\\&\le \Lambda _{u_0}(|x-z|) +T \Lambda _{f}(|x-z|) + 3{\overline{\Lambda }}_{u_0,f}(|{\tilde{t}}-t|). \end{aligned} \end{aligned}$$

$\square$

By Arzelà-Ascoli, we obtain as a corollary that, up to a subsequence, the numerical solution converges locally uniformly to a limit.

Corollary 3.11

Assume the hypotheses of Proposition 3.10. Let $\{U_h\}_{h>0}$ be a sequence of solutions of (3.9). Then, there exist a subsequence $\{U_{h_l}\}_{l=1}^\infty$ and a function $u\in C_b({\mathbb {R}}^d\times [0,T])$ such that

$$\begin{aligned} U_{h_l}\rightarrow u \quad as \quad l\rightarrow \infty \quad locally uniformly in {\mathbb {R}}^N\times [0,T] . \end{aligned}$$

4 Convergence of the numerical scheme

From Corollary 3.11, we have that the sequence of numerical solutions has a subsequence converging locally uniformly to some function v. We will now show that v is a viscosity solution of (1.1).

Theorem 4.1

Let the assumptions of Corollary 3.11 hold. Then v is a viscosity solution of (1.1).

Proof

For notational simplicity, we avoid the subindex j and consider

$$\begin{aligned} U_{h}\rightarrow u \quad as \quad h\rightarrow 0 \quad locally uniformly in {\mathbb {R}}^N\times [0,T] . \end{aligned}$$

First of all, by the local uniform convergence,

$$\begin{aligned} u(x,0)=\lim _{h\rightarrow 0} U_h(x,0)=u_0(x), \end{aligned}$$

locally uniformly. We will now show that u is a viscosity supersolution. The proof that u is a viscosity subsolution is similar.

Now let $\varphi$ be a suitable test function for u at $(x^*,t^*)\in {\mathbb {R}}^d\times (0,T)$. We may assume that $\varphi$ satisfies

• (i) $\varphi (x^*,t^*)=u(x^*,t^* )$,

• (ii) $u(x,t)>\displaystyle \varphi (x,t)$ for all $(x,t)\in B_R(x^*)\times (t^*-R,t^*]\setminus \left( x^*,t^*\right)$.

The local uniform convergence ensures (see Section 10.1.1 in [17]) that there exists a sequence $\{(x^h,t^h)\}_{h>0}$ such that

• (i) $\varphi (x^h,t^h)-U_h(x^h,t^h)=\sup _{(x,t)\in B_R(x^h)\times (t^h-R,t^h]}\{\varphi (x,t)-U_h(x,t)\}=:M_h$,

• (ii) $\varphi (x^h,t^h)-U_h(x^h,t^h)\ge \displaystyle \varphi (x,t)-U_h(x,t)$ for all $(x,t)\in B_R(x^h)\times (t^h-R,t^h]\setminus (x^h,t^h)$

and

$$\begin{aligned} (x^h,t^h)\rightarrow (x^*,t^*) \quad as \quad h\rightarrow 0. \end{aligned}$$

Now consider $t_j\in {\mathcal {T}}_{\tau }$ such that $t^h\in [t_j,t_{j+1}]$ (note that the index j might depend on h, but this fact plays no role in the proof). By Remark 3.9,

$$\begin{aligned} U_h(x^h,t^h)= & {} U_h(x^h,t_j)+ (t^h-t_j) \sum _{y_\beta \in {\mathcal {G}}_h} J_p(U_h(x^h+y_\beta ,t_j)-U_h(x^h,t_j))\omega _\beta \\{} & {} \quad + (t^h-t_j) f(x^h). \end{aligned}$$

Define ${\widetilde{U}}_h=U_h+M_h$. It is clear that

$$\begin{aligned} {\widetilde{U}}_h(x^h,t^h)= & {} {\widetilde{U}}_h(x^h,t_j)+ (t^h-t_j) \sum _{y_\beta \in {\mathcal {G}}_h} J_p({\widetilde{U}}_h(x^h+y_\beta ,t_j)-{\widetilde{U}}_h(x^h,t_j))\omega _\beta \\{} & {} \quad +(t^h-t_j) f(x^h). \end{aligned}$$

Clearly, ${\widetilde{U}}_h(x^h,t^h)=\varphi (x^h,t^h)$ and ${\widetilde{U}}_h\ge \varphi$, which implies that

$$\begin{aligned} \varphi (x^h,t^h)= & {} {\widetilde{U}}_h(x^h,t_j)+ (t^h-t_j) \sum _{y_\beta \in {\mathcal {G}}_h} J_p({\widetilde{U}}_h(x^h+y_\beta ,t_j)-{\widetilde{U}}_h(x^h,t_j))\omega _\beta \nonumber \\{} & {} \quad +(t^h-t_j) f(x^h). \end{aligned}$$

(4.1)

Now consider the function $g:{\mathbb {R}}\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} g(\xi )=\xi +(t^h-t_j) \sum _{y_\beta \in {\mathcal {G}}_h} J_p({\widetilde{U}}(x^h+y_\beta ,t_j)-\xi )\omega _\beta \end{aligned}$$

and note that

$$\begin{aligned} g'(\xi )=1-(t^h-t_j) (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} |{\widetilde{U}}(x^h+y_\beta ,t_j)-\xi |^{p-2}\omega _\beta . \end{aligned}$$

We will check now that $g'(\xi )\ge 0$ for any $\xi \in [\varphi (x^h,t_j), {\widetilde{U}}(x^h,t_j)]$. Indeed,

$$\begin{aligned} \begin{aligned} |{\widetilde{U}}(&x^h+y_\beta ,t_j)-\xi | \le |{\widetilde{U}}(x^h+y_\beta ,t_j)-{\widetilde{U}}(x^h,t_j)|+|{\widetilde{U}}(x^h,t_j)-\xi |\\&\le |U(x^h+y_\beta ,t_j)-U(x^h,t_j)|+|{\widetilde{U}}(x^h,t_j)-\varphi (x^h,t_j)|\\&\le |U(x^h+y_\beta ,t_j)-U(x^h,t_j)|+|U(x^h,t_j)-U(x^h,t^h)|+|\varphi (x^h,t^h)-\varphi (x^h,t_j)|\\&\le \Lambda _{u_0}(|y_\beta |) +T \Lambda _{f}(|y_\beta |) + 3 {\overline{\Lambda }}_{u_0,f}(|t^h-t_j|)+|t^h-t_j| \Vert \partial _t \varphi \Vert _{L^\infty (B_R(x^h)\times [t^h-R, t^h])}\\&\le \Lambda _{u_0}(|y_\beta |) +T \Lambda _{f}(|y_\beta |) + 3 {\overline{\Lambda }}_{u_0,f}(\tau )+\tau \Vert \partial _t \varphi \Vert _{L^\infty (B_R(x^h)\times [t^h-R, t^h])}, \end{aligned} \end{aligned}$$

where we have used that ${\widetilde{U}}(x^h,t^h)=\varphi (x^h,t^h)$, Proposition 3.10 and the fact that $|t^h-t_j|\le \tau$. By ($CFL$), and taking $\tau$ small enough, we have

$$\begin{aligned} \begin{aligned} 3 {\overline{\Lambda }}_{u_0,f}(\tau )&+\tau \Vert \partial _t \varphi \Vert _{L^\infty (B_R(x^h)\times [t^h-R, t^h])}\\&\quad \le 3{\widetilde{K}} \tau ^{\frac{a}{2+(1-a)(p-2)}} + \left( 3\Vert f\Vert _{L^\infty ({\mathbb {R}}^d)} + \Vert \partial _t \varphi \Vert _{L^\infty (B_R(x^h)\times [t^h-R, t^h])}\right) \tau \\&\quad \le (3{\widetilde{K}}+1)\tau ^{\frac{a}{2+(1-a)(p-2)}}\\&\quad \le (3{\widetilde{K}}+1) r^a. \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} g'(\xi )&\ge 1-(t^h-t_j) (p-1) \sum _{y_\beta \in {\mathcal {G}}_h} | \Lambda _{u_0}(|y_\beta |) +T \Lambda _{f}(|y_\beta |) + (3{\widetilde{K}}+1) r^a|^{p-2}\omega _\beta \\&\ge 1-\tau (p-1) ( L_{u_0}+T L_f + 3{\widetilde{K}}+1)^{p-2}r^{a(p-2)}\sum _{y_\beta \in {\mathcal {G}}_h}\omega _\beta \\&\ge 1-\tau \frac{M(p-1) ( L_{u_0} +T L_{f} + 3{\widetilde{K}}+1)^{p-2}}{ r^{2+(1-a)(p-2)}}\\&\ge 0, \end{aligned} \end{aligned}$$

where we have used ($A _\omega$) and where the last inequality is due to the ($CFL$) condition. We can use this fact in (4.1) to get

$$\begin{aligned} \begin{aligned} \varphi (x^h,t^h)&= {\widetilde{U}}_h(x^h,t_j)+ (t^h-t_j) \sum _{y_\beta \in {\mathcal {G}}_h} J_p({\widetilde{U}}_h(x^h+y_\beta ,t_j)-{\widetilde{U}}_h(x^h,t_j))\omega _\beta +(t^h-t_j) f(x^h)\\&\ge \varphi (x^h,t_j)+ (t^h-t_j) \sum _{y_\beta \in {\mathcal {G}}_h} J_p({\widetilde{U}}_h(x^h+y_\beta ,t_j)-\varphi (x^h,t_j))\omega _\beta +(t^h-t_j) f(x^h)\\&\ge \varphi (x^h,t_j)+ (t^h-t_j) \sum _{y_\beta \in {\mathcal {G}}_h} J_p(\varphi (x^h+y_\beta ,t_j)-\varphi (x^h,t_j))\omega _\beta +(t^h-t_j) f(x^h). \end{aligned} \end{aligned}$$

Consistency ($A _{c}$) yields

$$\begin{aligned} \partial _t\varphi (x^h,t^h) + o(\tau )\ge \Delta _p\varphi (x^h,t_j)+ o_h(1)+f(x^h). \end{aligned}$$

Passing to the limit as $h,\tau \rightarrow 0$, we get the desired result by the regularity of $\varphi$ and the fact that $t^h,t_j\rightarrow t^*$ and $x^h\rightarrow x^*$ as $h\rightarrow 0$. $\square$

We are now ready to prove convergence of the scheme.

Proof of Theorem 2.2

By Corollary 3.11 and Theorem 4.1, we know that, up to a subsequence, the sequence $U_h$ converges to a viscosity solution of (1.1). Moreover, since viscosity solutions are unique (cf. Theorem 2.4), the whole sequence converges to the same limit.

5 Discretizations

In this section, we present two examples of discretizations and verify that the assumptions ($A _{c}$) and ($A _\omega$) are satisfied. Moreover, we also give the precise form of corresponding CFL-condition.

5.1 Discretization in dimension $d=1$

We consider the following finite difference discretization of $\Delta _p$ in dimension $d=1$

$$\begin{aligned} D_p^h \phi (x) = \frac{J_p(\phi (x+h)-\phi (x))+J_p(\phi (x-h)-\phi (x))}{h^p}. \end{aligned}$$

A proof of consistency ($A _{c}$) can be found in Theorem 2.1 in [8]. Assumption ($A _\omega$) is trivially true for $r=h$ since

$$\begin{aligned} \omega _1=\omega _{-1}= \frac{1}{h^p} \quad and \quad \omega _\beta =0 \quad otherwise, \end{aligned}$$

so that

$$\begin{aligned} \sum _{y_\beta \in {\mathcal {G}}_h} \omega _\beta =\frac{2}{h^p}. \end{aligned}$$

5.2 Discretization in dimension $d>1$

The following discretization was introduced in [9]:

$$\begin{aligned} D_p^h\phi (x)= \frac{h^d}{{\mathcal {D}}_{d,p}\, \omega _d\, r^{p+d}} \sum _{y_\beta \in B_r} J_p(\phi (x+y_\beta )-\phi (x)), \end{aligned}$$

where $\omega _d$ denotes the measure of the unit ball in ${\mathbb {R}}^d$, the relation between r and h is given by

$$\begin{aligned} h={\left\{ \begin{array}{ll} o\big (r^\frac{p}{p-1}\big ), &{} \quad if \quad p \in (2,3],\\ o\big (r^{\frac{3}{2}}\big ),&{} \quad if \quad p\in (3,\infty ),\\ \end{array}\right. } \end{aligned}$$

(5.1)

and . When $p\in {\mathbb {N}}$, a more explicit value of this constant is given in [9]. In general, the explicit value is given by

$$\begin{aligned} {\mathcal {D}}_{d,p} = \frac{d}{4\sqrt{\pi }}\cdot \frac{p-1}{d+p}\cdot \frac{\Gamma (\frac{d}{2})\Gamma (\frac{p-1}{2})}{\Gamma (\frac{d+p}{2})}. \end{aligned}$$

A proof of consistency ($A _{c}$) can be found in Theorem 1.1 in [9]. Assumption ($A _\omega$) trivially holds for $h=o(r^\alpha )$ for some $\alpha >0$ according to (5.1) since

$$\begin{aligned} \omega _\beta =\omega _{-\beta }= \frac{h^d}{{\mathcal {D}}_{d,p}\, \omega _d\, r^{p+d}} \quad if \quad |h\beta |<r \quad and \quad \omega _\beta =0 \quad otherwise . \end{aligned}$$

To check ($A _\omega$) we rely on the following estimate given in the proof of Theorem 1.1 in [9]:

$$\begin{aligned} \sum _{y_\beta \in B_r}h^d \le |B_{r+\sqrt{d}h}|. \end{aligned}$$

In particular, taking for example $h\le r/\sqrt{d}$, we have

$$\begin{aligned} \sum _{y_\beta \in B_r} \omega _\beta = \frac{1}{{\mathcal {D}}_{d,p}r^{p}}\frac{|B_{r+\sqrt{d}h}|}{|B_{r}|}\le \frac{2^d}{{\mathcal {D}}_{d,p}r^{p}}. \end{aligned}$$

6 Numerical experiments

We will perform the numerical tests comparing the numerical solution with the explicit Barenblatt solution of (1.1). For $p>2$ this is given by

$$\begin{aligned} B(x,t)=K t^{-\alpha }\left( 1- \left( \frac{|x|}{t^\beta }\right) ^{\frac{p}{p-1}}\right) ^{\frac{p-1}{p-2}}_+, \end{aligned}$$

where the constants are,

$$\begin{aligned} \alpha =\frac{d}{d(p-2)+p}, \quad \beta =\frac{1}{d(p-2)+p}, \quad and \quad K=\left( \frac{p-2}{p}\beta ^{\frac{1}{p-1}}\right) ^{{\frac{p-1}{p-2}}}. \end{aligned}$$

6.1 Simulations in dimension $d=1$

We consider the initial condition

$$\begin{aligned} u_0(x)=B(x,1)=K\left( 1-|x|^{\frac{p}{p-1}}\right) ^{\frac{p-1}{p-2}} \end{aligned}$$

and $f=0$. The corresponding solution of problem (1.1) is given by (see [24])

$$\begin{aligned} u(x,t)=B(x,t+1)=K (t+1)^{-\alpha }\left( 1- \left( \frac{|x|}{(t+1)^\beta }\right) ^{\frac{p}{p-1}}\right) ^{\frac{p-1}{p-2}}_+ . \end{aligned}$$

Let us now comment on the CFL-condition ($CFL$). Clearly, $u_0$ is a Lipschitz function, and we can give an upper bound to its Lipschitz constant as follows

$$\begin{aligned} L_{u_0}=\sup _{x\in [-1,1]} \left| \frac{du_0}{dx}(x)\right| =\sup _{r\in [0,1]}\left\{ K \frac{p}{p-2}\left( 1-r^{\frac{p}{p-1}}\right) ^{\frac{1}{p-2}}r^{\frac{1}{p-1}}\right\} \le K \frac{p}{p-2}. \end{aligned}$$

Thus, for all $p>2$, the CFL condition ($CFL$) can be take as $\tau \sim h^2$ (since $f=0$ in this case). For completeness, we find the value of K in dimension $d=1$. Note that

$$\begin{aligned} K=\left( \frac{p-2}{p}\frac{1}{(2(p-1))^{\frac{1}{p-1}}}\right) ^{{\frac{p-1}{p-2}}}=\left( \frac{p-2}{p}\right) ^{{\frac{p-1}{p-2}}}\frac{1}{(2(p-1))^{\frac{1}{p-2}}}, \end{aligned}$$

so that $L_{u_0}\le \left( \frac{p-2}{2p(p-1)}\right) ^{{\frac{1}{p-2}}}.$

In Fig. 1, we show the numerical errors obtained. As it can be seen there, the errors seem to behave like $O(h^{p/(p-1)})$.

Fig. 1

Errors in dimension $d=1$ for $p=3,4,10,100$

Appendices

Appendix A: Estimates for mollified Hölder continuous functions

Here we present some explicit estimates for mollifications needed in the proof of equicontinuity in Lemma 3.7. Let $\tau :[0,\infty )\rightarrow {\mathbb {R}}$ be a smooth function such that ${{\,\textrm{supp}\,}}\tau \subset [0,1 ]$. Define $\rho :{\mathbb {R}}^d\rightarrow {\mathbb {R}}$ given by $\rho (x)=\frac{M}{\omega _d}\tau (|x|)$ where $\omega _d$ is the measure of the unit sphere in dimension d and $M=M(d)$ is a constant defined by

$$\begin{aligned} M=\left( \int _0^1 \tau (r) r^{d-1}\,\textrm{d}r\right) ^{-1}. \end{aligned}$$

In this way, we have that $\int _{B_1}\rho (x)\,dx = 1$. For $\delta >0$ define also

$$\begin{aligned} \rho _\delta (x)=\frac{1}{\delta ^d}\rho \left( \frac{x}{\delta }\right) . \end{aligned}$$

Then, for a function $f\in L^1_{\text {loc}}$ we define the mollification of f as

$$\begin{aligned} f_\delta (x)=(f*\rho _\delta )(x)=\int _{B_\delta }\rho _\delta (y)f(x-y)\,\textrm{d}y =\int _{{\mathbb {R}}^n}\rho _\delta (x-y)f(y) \,\textrm{d}y. \end{aligned}$$

The lemma below gives an estimate of the Lipschitz seminorm of $f_\delta$ when f is an $\alpha$-Hölder continuous function for some $\alpha \in (0,1]$.

Lemma A.1

Let $\alpha \in (0,1]$. Consider a function $f\in C^\alpha ({\mathbb {R}}^n)$ with $|f(x)-f(y)|\le L|x-y|^\alpha$ for all $x,y\in {\mathbb {R}}^d$. Then, for all $x,y\in {\mathbb {R}}^d$, we have that

$$\begin{aligned}|f_\delta (x)-f_\delta (y)|\le K_1 L|x-y|\delta ^{\alpha -1}, \quad with \quad K_1= M \int _0^1|\tau '(r)| r^{d-1}\,\textrm{d}r. \end{aligned}$$

Proof

Since $\int _{{\mathbb {R}}^n}\nabla \rho _\delta (y)\,dy=\int _{{\mathbb {R}}^n}\nabla \rho (y)\,dy= 0$, it follows that

$$\begin{aligned} |\nabla f_\delta (x)|{} & {} =\left| \int _{B_\delta }\nabla \rho _\delta (y)\left( f(x-y)-f(x)\right) \,\textrm{d}y\right| \le L\int _{B_\delta }\left| \nabla \rho _\delta (y)\right| |y|^\alpha \,\textrm{d}y \nonumber \\{} & {} \le \frac{M}{\omega _d} L \delta ^\alpha \int _{B_\delta }\frac{1}{\delta ^{d+1}}\left| \tau '\left( \frac{|y|}{\delta }\right) \right| \,\textrm{d}y=M L \delta ^{\alpha -1} \int _0^1|\tau '(r)| r^{d-1}\,\textrm{d}r. \nonumber \\ \end{aligned}$$

(A.1)

Thus,

$$\begin{aligned} |f_\delta (x)-f_\delta (y)|\le \Vert \nabla f_\delta \Vert _{L^\infty ({\mathbb {R}}^d)}|x-y|=K_1L|x-y|\delta ^{\alpha -1}. \end{aligned}$$

$\square$

The lemma below gives an estimate of the second order central difference quotients of $f_\delta$ when f is an $\alpha$-Hölder continuous function for some $\alpha \in (0,1]$.

Lemma A.2

Let $\alpha \in (0,1]$. Consider a function $f\in C^\alpha ({\mathbb {R}}^n)$ with $|f(x)-f(y)|\le L|x-y|^\alpha$ for all $x,y\in {\mathbb {R}}^d$. Then, for all $x,y\in {\mathbb {R}}^d$, we have that

$$\begin{aligned} |f_\delta (x+y)+f_\delta (x-y)-2f_\delta (x)|\le K_2L|y|^2\delta ^{\alpha -2}, \end{aligned}$$

with

$$\begin{aligned} K_2= M\int _0^1\left( \frac{|\tau '(r)|}{r} +|\tau ''(r)|\right) r^{d-1}\,\textrm{d}r . \end{aligned}$$

Proof

We note that we have the following formula for the second order derivatives of $\rho _\delta$:

$$\begin{aligned} \partial _{ij} \rho _\delta (y)=\left( \frac{\delta _{ij}}{|y|}-\frac{y_i y_j}{|y|^3}\right) \frac{1}{\delta ^{d+1}}\tau '\left( \frac{|y|}{\delta }\right) +\frac{y_i y_j}{\delta ^{d+2}|y|^2}\tau ''\left( \frac{|y|}{\delta }\right) , \end{aligned}$$

so that

$$\begin{aligned} \langle D^2 \rho _\delta (y) \xi ,\xi \rangle \le \left( \frac{1}{\delta ^{d+1}|y|}\left| \tau '\left( \frac{|y|}{\delta }\right) \right| +\frac{1}{\delta ^{d+2}}\left| \tau ''\left( \frac{|y|}{\delta }\right) \right| \right) |\xi |^2. \end{aligned}$$

Similarly to the gradient, the Hessian also integrates to zero, that is,

$$\begin{aligned} \int _{{\mathbb {R}}^n}\partial _{ij} \rho _\delta (y)\,\,\textrm{d}y = 0\quad for all \quad i,j=1,\ldots ,d. \end{aligned}$$

Indeed, when $i\ne j$, the result follows by antisymmetry in y. When $i=j$, we are integrating $\partial _{ii} \rho _\delta$, which yields zero since $\partial _i \rho _\delta =0$ on ${\mathbb {R}}^d \setminus B_\delta$. As in the proof of the previous lemma, it follows that

$$\begin{aligned} \Vert D^2 f_\delta \Vert{} & {} =\left\| \int _{B_\delta }D^2 \rho _\delta (y)\left( f(x-y)-f(x)\right) dy\right\| \le L\int _{B_\delta }\Vert D^2 \rho _\delta (y)\Vert |y|^\alpha \,\textrm{d}y \nonumber \\{} & {} \le \frac{M}{\omega _d} L \delta ^{\alpha } \int _{B_\delta }\left( \frac{1}{\delta ^{d+1}|y|}\left| \tau '\left( \frac{|y|}{\delta }\right) \right| +\frac{1}{\delta ^{d+2}}\left| \tau ''\left( \frac{|y|}{\delta }\right) \right| \right) \,\textrm{d}y \nonumber \\{} & {} =ML \delta ^{\alpha -2} \int _0^1\left( \frac{|\tau '(r)|}{r} +|\tau ''(r)|\right) r^{d-1}\,\textrm{d}r, \nonumber \\ \end{aligned}$$

(A.2)

where $\Vert \cdot \Vert$ denotes the operator norm. Now, by Taylor expansion

$$\begin{aligned} f_\delta (x \pm y)=f_\delta \phi (x)\pm \nabla f_\delta (x)\cdot y+ \frac{1}{2}\sum _{\beta =1} \frac{\partial ^{|\beta |}f_\delta }{\partial x_\beta }( z^{\pm }) y^\beta . \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} |f_\delta (x + y)+f_\delta (x - y)-2f_\delta (x)|&\le |y|^2 \Vert D^2 f_\delta (z) \Vert \\&\le M \left( \int _0^1\left( \frac{|\tau '(r)|}{r} +|\tau ''(r)|\right) r^{d-1}\,\textrm{d}r \right) L \delta ^{\alpha -2}|y|^2. \end{aligned} \end{aligned}$$

$\square$

1.1 A.1. Explicit constants in dimensions one, two and three

Here we will compute explicit constants for the mollifier that is based on the choice

$$\begin{aligned} \tau (r)= e^{-\frac{1}{1-r^2}}\chi _{ [0,1)}(r). \end{aligned}$$

In one dimension: We have

$$\begin{aligned} M = \left( \int _0^1 \tau (r) \,\textrm{d}r\right) ^{-1}\le 4.51 \end{aligned}$$

and

$$\begin{aligned} K_1=M\int _0^1 |\tau '(r)|\,\textrm{d}r = M\int _0^1(- \tau '(r)) \,\textrm{d}r=M \tau (0)=\frac{M}{e}\le 1.67. \end{aligned}$$

Since

$$\begin{aligned} \int _0^1 \frac{|\tau '(r)|}{r}=2\int _0^1 \frac{e^{-\frac{1}{1-r^2}}}{(1-r^2)^2}\,\textrm{d}r \le 0.8, \end{aligned}$$

and

$$\begin{aligned} \int _0^1 |\tau ''(r)|dr=\int _0^1 \left| e^{-\frac{1}{1-r^2}}\frac{6r^4-2}{(1-r^2)^4}\right| \,\textrm{d}r\le 1.6, \end{aligned}$$

we conclude that

$$\begin{aligned} K_2\le 2.4M\le 10.83. \end{aligned}$$

In two dimensions: We have

$$\begin{aligned} M = \left( \int _0^1 \tau (r) r\,\textrm{d}r\right) ^{-1}\le 13.47 \end{aligned}$$

and

$$\begin{aligned} K_1=M\int _0^1 |\tau '(r)|r\,\textrm{d}r = M\int _0^1(- \tau '(r)r) \,\textrm{d}r=M\int _0^1\tau (r) \,\textrm{d}r\le 0.23M\le 3.13. \end{aligned}$$

Since

$$\begin{aligned} \int _0^1 |\tau '(r)| \,\textrm{d}r=\frac{ 1}{e} \end{aligned}$$

and

$$\begin{aligned} \int _0^1 |\tau ''(r)| r dr=\int _0^1 \left| e^{-\frac{1}{1-r^2}}\frac{6r^4-2}{(1-r^2)^4}r \right| \,\textrm{d}r\le 1.04, \end{aligned}$$

we conclude that

$$\begin{aligned} K_2\le M(e^{-1}+1.04)\le 18.97. \end{aligned}$$

In three dimensions: We have

$$\begin{aligned} M = \left( \int _0^1 \tau (r) r^2\,\textrm{d}r\right) ^{-1}\le 28.49 \end{aligned}$$

and

$$\begin{aligned} K_1=M\int _0^1 |\tau '(r)|r^2\,\textrm{d}r = M\int _0^1(- \tau '(r)r^2) \,\textrm{d}r=2M\int _0^1\tau (r)r \,\textrm{d}r\le 2\times 0.08 M\le 4.56. \end{aligned}$$

Since

$$\begin{aligned} \int _0^1 |\tau '(r)|r\,\textrm{d}r=\int _0^1\tau (r) \,\textrm{d}r\le 0.23 \end{aligned}$$

and

$$\begin{aligned} \int _0^1 |\tau ''(r)| r^2 dr=\int _0^1 \left| e^{-\frac{1}{1-r^2}}\frac{6r^4-2}{(1-r^2)^4}r^2 \right| \,\textrm{d}r\le 0.79, \end{aligned}$$

we conclude that

$$\begin{aligned} K_2\le M(0.23+0.79)\le 29.06. \end{aligned}$$

Appendix B: Pointwise inequalities

The following lemma follows from the Taylor expansion of the function $t\mapsto |t|^{p-2}t$.

Lemma B.1

Let $p\ge 2$. Then

$$\begin{aligned} \Big ||a+b|^{p-2}(a+b)-|a|^{p-2}a\Big |\le (p-1)\max ( |a|,|a+b|)^{p-2}|b|. \end{aligned}$$