Systems of Finite Difference Equations, Numerical Solutions
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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- Weinberger, H., Variational Methods for Eigenvalue Approximation, Regional Conference Series in Applied Mathematics, SIAM, 15, Philadelphia, Pa., 1974.Google Scholar
- Forsythe, G. E. and Wasow, W. R., Finite Difference Methods for Partial Differential Equations, Wiley, N. Y., 1960.Google Scholar
- Ames, W. F., Numerical Methods for Partial Differential Equations, second ed., Acadmic Press, N. Y., Thomas Nelson & Sons, London, 1977.Google Scholar
- Birkhoff, G. and Lynch, R. E., Numerical Solution of Elliptic Problems, SIAM Studies in Applied Math., 6, 1984.Google Scholar