Wave propagation

From CFD-Wiki

Contents 1 Introduction 2 Governing Equation 3 Initial Condition 4 Exact Solution 5 Compact scheme 6 Runge-Kutta 7 Sample result 8 Reference

Introduction

The 1-D wave propagation problem is used as a test case for studying the dissipation and dispersion errors in a given finite difference scheme.There are a number of schemes which can be used to solve the problem. The usage of a higher order compact stencil (4th order considered here) with a low storage 4th order Runge-Kutta scheme to solve the current problem is discussed.

Governing Equation

Initial Condition

Exact Solution

Compact scheme

At Boundaries

Runge-Kutta

Consider

The low storage scheme is implemented as follows

where M refers to the stages ,dt is the time step and the coefficients a and b are given by

a[5]={0,-0.41789047,-1.19215169,-1.69778469,-1.51418344}

b[5]={0.149665602,0.37921031,0.82295502,0.69945045,0.15305724}

Sample result

Reference

• Williamson, Williamson (1980), "Low Storage Runge-Kutta Schemes", Journal of Computational Physics, Vol.35, pp.48–56.

• Lele, Lele, S. K. (1992), "Compact Finite Difference Schemes with Spectral-like Resolution,” Journal of Computational Physics", Journal of Computational Physics, Vol. 103, pp 16–42.