Pade approximations in the numerical solution of hyperbolic differential equations

  • R. A. Williamson
Conference paper
First Online:
Part of the Lecture Notes in Mathematics book series (LNM, volume 1071)

Abstract

The relationship between the accuracy and stability of a semidiscretised finite difference scheme for solving the advection equation, ut = ux, 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 ZLlnz are normal. Furthermore, using Padé theory we find stable methods attaining the bound on accuracy for various choices of r,s,R and S.

Keywords

Rational Approximation Stable Approximation Error Constant Interpolation Point Efficient Pole 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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5. References

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Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • R. A. Williamson
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsCambridgeEngland

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