# Coupled 1d partial differential equations

I have been trying to numerically solve the 1d homogeneous wave equation

\begin{align} u_{tt}(x,t) &=& u_{xx}(x,t) \\ u(x,0) &=& f(x) \\ u_t(x, 0) &=& 0 \\ u(0, t) &=& u(1, t) = 0 \end{align}

by casting it into two first-order linear equations

\begin{align} u_t &=& v_x \\ v_t &=& u_x \end{align}

This is what I've tried

1. Apply forward finite differences for the $t$ derivatives and centered differences for the $x$ derivatives

\begin{align} u^{n+1}_j &=& u^n_j + \frac{\alpha}{2}(v^n_{j + 1} - v^n_{j - 1}) \\ v^{n+1}_j &=& v^n_j + \frac{\alpha}{2}(u^n_{j + 1} - u^n_{j - 1}) \end{align}

where $u^n_j = u(t_n, x_j)$ and $\alpha = \Delta t/\Delta x < 1$. As initial condition $f(x)$ I used a gaussian function of variance 0.1. This is what I get when the pulse approaches the boundaries: Colors label different times, until the wave hits the boundaries, were it gets really bad!

1. I also followed this document

Lecture notes from Prof. L. Rezzolla

Turns out he introduces yet another variable I don't see the need for. See his Chapter 5. It sort of works here, but still the wave gets really distorted when it reaches the edges of the $x$-domain

1. Upgraded the order of the finite differences for the $x$-variable, it does not improve.

I am out of ideas, any suggestion is much appreciated

• Please describe, how you deal with $v_0^{n+1}$ and $v_J^{n+1}$ (boundary values), that's crucial – uranix Nov 1 '16 at 21:57

$\frac{u^{n+1}_j - 2u^n_j + u^{n-1}_j}{\Delta t^2} = \frac{u^n_{j+1} - 2u^n_j + u^n_{j-1}}{\Delta x^2}.$
If you do a von Neumann analysis, the stability criteria is $\Delta t \leq \Delta x$. This should give you no issues near the boundary.
If you want to see why your scheme is unstable, do a von Neumann analysis. The way to do it here is to plug $v^n_{j+1}$ and $v^n_{j-1}$ into the scheme for $u^{n+1}_j$. After a lot of simplification you will get a scheme in terms of only $u$. Then do the usual von Neumann analysis: Look for a separable solution $u^n_j =\lambda_k^n e^{ikj\Delta x}$ and find $\lambda_k$. It should turn out that no mater the grid resolution, at least one eiegnvalue $\lambda_k$ has modulus larger than 1, so the scheme is unstable.