1 Introduction

Dynamic iterations have been broadly investigated as numerical methods applied to solve differential systems on parallel computers and are often called waveform relaxation techniques. These techniques have been introduced by Lelarasmee et al. [4] and investigated by many authors for different kinds of differential equations, see, for example, [3] and [5] for systems of ordinary differential equations, [1] and [2] for systems of delay differential equations and [6] and [7] for general functional differential equations. However, these techniques have been mainly investigated in the context of parallel computations and not much attention has been given to answer the question of whether or not permutations of the equations in a given system influence the convergence of the applied dynamic iterations. Recent investigations [8] in this direction for linear systems of differential equations show that appropriately chosen permutations, in light of the values of the model parameters, present a way to speed up the convergence of dynamic iterations. The goal of the current paper is to address this question for nonlinear differential equations.

In this paper, we investigate dynamic iterations for systems written in the form

$$\displaystyle \begin{aligned} \left\{ \begin{array}{lcr} \displaystyle \frac{d}{dt}x=ax+xf_1(y)+g_1(t)\\[0.3cm] \displaystyle \frac{d}{dt}y=\tilde{a}y+yf_2(x)+g_2(t) \end{array} \right. \end{aligned} $$
(1)

where a, \(\tilde {a}\) are real parameters and f i, g i, i = 1, 2, are given real functions. The system (1) is supplemented by the initial conditions

$$\displaystyle \begin{aligned}x(0)=\xi_0,\quad y(0)=\eta_0.\end{aligned} $$

For an arbitrary continuous function y (0)(t), we consider the sequences \(\{ x^{(k)}(t)\}_{k=1}^{\infty }\), \(\{ y^{(k)}(t)\}_{k=0}^{\infty }\) defined by

$$\displaystyle \begin{aligned} \left\{ \begin{array}{lcr} \displaystyle \frac{d}{dt}x^{(k+1)}=ax^{(k+1)}+x^{(k+1)}f_1(y^{(k)})+g_1(t),\\[0.3cm] \displaystyle \frac{d}{dt}y^{(k+1)}=\tilde{a}y^{(k+1)}+y^{(k+1)}f_2(x^{(k+1)})+g_2(t), \end{array} \right. \end{aligned} $$
(2)

where k = 0, 1, 2, … and

$$\displaystyle \begin{aligned}x^{(k+1)}(0)=\xi_0, \quad y^{(k+1)}(0)=\eta_0.\end{aligned} $$

The numerical scheme (2) is called Gauss-Seidel waveform relaxation. The advantage of (2) over (1) is that the application of implicit methods, for example BDF methods, for integration of (2) in time t does not require solving nonlinear algebraic equations at each time step. Note that the application of implicit time integration methods to the nonlinear differential system (1) leads to a system of nonlinear algebraic equations that require an additional process to solve them at each time step (more time steps mean that more nonlinear algebraic systems need to be solved). Such an additional process would not be needed if system (2) would be applied.

The paper is organized as follows. In Sect. 2, we analyze the convergence of the sequence \(\{ (x^{(k)}(t), y^{(k)}(t))\}_{k=0}^{\infty }\) to the exact solution (x(t), y(t)) as k →∞. Then, in Sect. 3, we present results of numerical experiments involving nonlinear systems applied in population dynamics. Finally, we finish with concluding remarks in Sect. 4.

2 Error Analysis

In this section, we analyze the errors

$$\displaystyle \begin{aligned}e_x^{(k)}(t)=x^{(k)}(t)-x(t), \quad e_y^{(k)}(t)=y^{(k)}(t)-y(t), \end{aligned}$$

as k →∞. We assume that the unknown solutions x, y are bounded and the given functions f 1, f 2 are Lipschitz continuous. In what follows, we use the following notation. Let L 1, L 2 be Lipschitz constants for f 1 and f 2, respectively; that is,

$$\displaystyle \begin{aligned}\begin{array}{l} |f_1(y)-f_1(\tilde{y})|\leq L_1|y-\tilde{y}|, \quad \text{for all } y,\tilde{y} \in \mathbb{R}, \\[0.1cm] |f_2(x)-f_2(\tilde{x})|\leq L_2|x-\tilde{x}|, \quad \text{for all } x,\tilde{x} \in \mathbb{R} \end{array} \end{aligned}$$

and X, Y , R, E 0 be positive constants such that

$$\displaystyle \begin{aligned}\begin{array}{l} |x(t)| \le X, \quad |y(t)| \leq Y, \quad \text{for all } 0\leq t,\\[0.1cm] f_1(y)+a\le R, \quad f_2(x)+\tilde{a}\leq R, \quad \text{on bounded sets}, \\[0.1cm] |e_y^{(0)}(t)|\leq E_0, \quad \mbox{for all } 0\leq t. \end{array} \end{aligned}$$

The following theorem provides error bounds for scheme (2).

Theorem 1

Let k = 0, 1, 2, … and t ≥ 0. Then,

$$\displaystyle \begin{aligned} |e_x^{(k+1)}(t)| &\le E_0 XL_1\big(XYL_1L_2\big)^k\frac{e^{Rt}}{R^{2k+1}}\sum_{j=2k+1}^{\infty}\frac{(-1)^{j+1}(Rt)^j}{j!}, \end{aligned} $$
(3)
$$\displaystyle \begin{aligned} |e_y^{(k+1)}(t)| & \le E_0\big(XYL_1L_2\big)^{k+1} \frac{e^{Rt}}{R^{2k+2}}\sum_{j=2k+2}^{\infty}\frac{(-1)^j(Rt)^j}{j!}. \end{aligned} $$
(4)

Proof

Note that \(e_x^{(k)}(0)=0\) and \(e_y^{(k)}(0)=0\), for all k = 0, 1, 2, …. Then, from (1) and (2), we get

$$\displaystyle \begin{aligned} |e_x^{(k+1)}(t)| & \le XL_1 \int_{0}^{t}|e_y^{(k)}(\tau )|e^{R(t-\tau )}d\tau, \end{aligned} $$
(5)
$$\displaystyle \begin{aligned} |e_y^{(k+1)}(t)| & \le YL_2 \int_{0}^{t}|e_x^{(k+1)}(\tau )|e^{R(t-\tau )}d\tau, \end{aligned} $$
(6)

for k = 0, 1, 2, … and t ≥ 0. From (5), we get

$$\displaystyle \begin{aligned} |e_x^{(1)}(t)|&\le XL_1E_0\int_{0}^{t}e^{R(t-\tau)} d\tau = \frac{XL_1E_0}{R}\Big( e^{Rt}-1\Big) = \frac{XL_1E_0}{R}e^{Rt}\Big( 1 - \sum_{j=0}^{\infty}\frac{(-Rt)^j}{j!}\Big)\\ &=\frac{XL_1E_0}{R}e^{Rt} \sum_{j=1}^{\infty}\frac{(-1)^{j+1}(Rt)^j}{j!}, \end{aligned}$$

which shows (3) for k = 0. We now use (6) and get

$$\displaystyle \begin{aligned} |e_y^{(1)}(t)|&\le \frac{XYL_1L_2E_0}{R}e^{Rt}\int_{0}^{t}\sum_{j=1}^{\infty}\frac{(-1)^{j+1}(R\tau)^j}{j!}d\tau \\ &=\frac{XYL_1L_2E_0}{R}e^{Rt} \sum_{j=1}^{\infty}\frac{(-1)^{j+1}R^jt^{j+1}}{(j+1)!} =\frac{XYL_1L_2E_0}{R^2}e^{Rt} \sum_{j=2}^{\infty}\frac{(-Rt)^{j}}{j!}, \end{aligned}$$

which shows (4) for k = 0. We now assume (3) and (4) for a certain k. Then, from (5) and (4), we get

$$\displaystyle \begin{aligned} |e_x^{(k+2)}(t)|&\le XL_1\int_{0}^{t} \big( XYL_1L_2\big)^{k+1}E_0\frac{e^{R\tau}}{R^{2k+2}} \sum_{j=2k+2}^{\infty}\frac{(-R\tau)^j}{j!}e^{R(t-\tau)}d\tau \\ &=XL_1\big( XYL_1L_2\big)^{k+1}E_0 \frac{e^{Rt}}{R^{2k+2}}\sum_{j=2k+2}^{\infty} \frac{(-R)^jt^{j+1}}{(j+1)!}\\ &=XL_1\big( XYL_1L_2\big)^{k+1}E_0 \frac{e^{Rt}}{R^{2k+3}}\sum_{j=2k+3}^{\infty} \frac{(-1)^{j+1}(Rt)^j}{j!}, \end{aligned}$$

which, by mathematical induction, shows (3). We now use (6) and (3) and obtain the following result,

$$\displaystyle \begin{aligned} |e_y^{(k+2)}(t)|&\le YL_2\int_{0}^{t} XL_1 \big(XYL_1L_2\big)^{k+1}E_0\frac{e^{R\tau }}{R^{2k+3}}\sum_{j=2k+3}^{\infty}\frac{(-1)^{j+1}(R\tau)^j}{j!}e^{R(t-\tau)}d\tau\\ &= \big(XYL_1L_2\big)^{k+2}E_0\frac{e^{Rt}}{R^{2k+4}} \sum_{j=2k+4}^{\infty} \frac{(-Rt)^j}{j!}, \end{aligned}$$

which shows (4) and finishes the proof. □

3 Numerical Experiments and Methods Comparison

In this section, we present results of numerical experiments involving dynamic iterations applied to Volterra equations for predator-prey interactions.

The system of interest is of the form

$$\displaystyle \begin{aligned} \left\{ \begin{array}{lcr} \displaystyle \frac{d}{dt}x=ax - bxy +g_1(t)\\[0.3cm] \displaystyle \frac{d}{dt}y=-cy+dxy+g_2(t) \end{array} \right. \end{aligned} $$
(7)

where a = 2∕3, b = 20, c = 50, d = 0.01, 0 ≤ t ≤ 10. We apply (2) and obtain the following scheme

$$\displaystyle \begin{aligned} \left\{ \begin{array}{lcr} \displaystyle \frac{d}{dt}x^{(k+1)}=ax^{(k+1)} - bx^{(k+1)}y^{(k)} +g_1(t)\\[0.3cm] \displaystyle \frac{d}{dt}y^{(k+1)}=-cy^{(k+1)}+dx^{(k+1)}y^{(k+1)}+g_2(t). \end{array} \right. \end{aligned} $$
(8)

If we write the equations in system (7) in the opposite order and then apply (2), we obtain a different scheme of the form

$$\displaystyle \begin{aligned} \left\{ \begin{array}{lcr} \displaystyle \frac{d}{dt}y^{(k+1)}=-cy^{(k+1)}+dx^{(k)}y^{(k+1)}+g_2(t)\\[0.3cm] \displaystyle \frac{d}{dt}x^{(k+1)}=ax^{(k+1)} - bx^{(k+1)}y^{(k+1)} +g_1(t). \end{array} \right. \end{aligned} $$
(9)

Numerical solutions x (k)(t n) and y (k)(t n) computed by (8) and (9) are presented in Fig. 1 as functions of t n for k = 6 in the case of (8) and for k = 4 in the case of (9).

Fig. 1
figure 1

Numerical solutions of (7): x(t) solid and y(t) dashed

Full size image

Although both schemes (8) and (9) originate from the same application of (2), their errors are different and demonstrate different convergence rates. The errors of both schemes are presented in Fig. 2. The upper subplot demonstrates the errors resulting from the application of (8) and the lower subplot demonstrates the errors resulting from the application of (9). Each scheme is integrated by BDF3 with h = 10−4.

Fig. 2
figure 2

Methods comparison: errors by (8) (upper subplot) and errors by (9) (lower subplot)

Full size image

We now compare the accuracy and CPU time using the solver ode15s and scheme (9) integrated by BDF6 with h = 10−2. The maximum error

$$\displaystyle \begin{aligned} \max \Big\{ \max_{n}\big|x_1(t_n)-x_{1,n}^{(k)}\big|, \max_{n}\big|x_2(t_n)-x_{2,n}^{(k)}\big| \Big\} \end{aligned}$$

is 7.03 ⋅ 10−13 and 1.91 ⋅ 10−14 using the solver ode15s and (9), respectively. The errors resulting from the application of both methods are comparable. The CPU time is 0.21 s when the solver ode15s is applied and it is 0.03 s when the numerical scheme (9) is applied, demonstrating that (9) is faster than ode15s.

4 Conclusions

In this work, we investigate dynamic iterations for nonlinear systems of differential equations applied in population dynamics. The advantage of dynamic iterations is that they allow to apply implicit time integration methods without the cost of solving nonlinear algebraic equations at each time step. We conclude that the convergence of dynamic iterations is different if we swap the order of the nonlinear differential equations in the given system even though the iterations are applied to the same system. That is, only by swapping the order of the equations, we can increase the rate of the convergence of the iterations. We also conclude that after choosing the optimal permutation of the equations, the proposed numerical scheme based on dynamical iterations is faster than the variable order method.