# Systems of Finite Difference Equations, Numerical Solutions

## Abstract

In this chapter we adapt the monotone schemes method to find approximate solutions for semilinear elliptic systems. We combine finite difference method with the monotone procedures developed in the last chapter. Accelerated version of the schemes is also considered in Section 6.3. We will consider up to two dimensional domain in Section 6.4, the method can naturally extend to higher dimensions. We will be only concerned with positive solutions to systems with Volterra-Lotka type ecological interactions. The method can however carry over to other interactions with similar monotone properties (c.f. Section 5.3). Further, the acceleration method can be applied to nonlinear interactions, with the appropriate convexity property (c.f. Equation (6.3-4)).

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