Pade approximations in the numerical solution of hyperbolic differential equations

Abstract

The relationship between the accuracy and stability of a semidiscretised finite difference scheme for solving the advection equation, u_t = u_x, is posed as an investigation into rational approximations to the logarithmic function. The geometric properties of such approximations are studied using the theory of order stars. In consequence we prove that the accuracy order p of the given stable scheme

$$\mathop \Sigma \limits_{j = - R}^S f_j \frac{d}{{dt}} U_{m + j} (t) = \frac{1}{{\Delta x}} \mathop \Sigma \limits_{j = - r}^S g_j U_{m + j} (t)$$

is bounded by min{r+s+R+S, 2(r+R+1), 2(s+S)}. We also demonstrate that particular Padé approximations to Z^Llnz are normal. Furthermore, using Padé theory we find stable methods attaining the bound on accuracy for various choices of r,s,R and S.