# Wave propagation

## 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

$\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x}=0$

## Initial Condition

$u(x,0)=exp(-b*(x-xc)^2)$

## Exact Solution

$u(x,t)=exp(-b*((x-ct)-xc)^2)$

## Compact scheme

${f_{j-1}}^{'}-4{f_j}^{'}+{f_{j+1}}^{'}=\frac{3}{h}(f_{j+1}-f_{j-1})+0(h^4)$

At Boundaries

${f_0}^{'}+2{f_1}^{'}=\frac{1}{h}(-2.5f_0+2f_1+0.5f_2)$
${f_n}^{'}+2{f_{n-1}}^{'}=\frac{1}{h}(2.5f_n-2f_{n-1}-0.5f_{n-2})$

## Runge-Kutta

Consider

$\frac {\partial U}{\partial t}=H$

The low storage scheme is implemented as follows

$U^{M+1}=U^M+b^Mdtf^M$
$f^M=a^Mf^{M-1}+H$

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}

## 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.