Exact finite-difference schemes for two-dimensional linear systems with constant coefficients. (English) Zbl 1151.65066

The autor presents an exact finite-difference method that can be applied to any two-dimensional linear system of first-order differential equations with constant coefficients. Table 1 gives the six possible cases of an exact finite-difference method \[ (1/\varphi)(x_{k+1}-x_k)=A(\theta x_{k+1}+(1-\theta)x_k) \] for the differential system \(x'(t)=A x(t)\), where the functions \(\varphi\) and \(\theta\) are determined by the two eigenvalues of \(A\), the step size \(h\) and each of the six Jordan canonical forms of \(A\). There is an alternative method in one case, and three special cases in another case of the Jordan canonical form.


65L12 Finite difference and finite volume methods for ordinary differential equations
34A30 Linear ordinary differential equations and systems
65L05 Numerical methods for initial value problems involving ordinary differential equations
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