## TestMoL

Roland Haas <rhaas@caltech.edu>

February 12 2014

This thorn provides tests (partially correctness tests) for MoL. It integrates a known function in
time choosing for each ODE method a polynomial in time that can be exactly integrated by the
method.

### 1 Introduction

The method of lines thorn (MoL) provides time integration facilities in Cactus. It basically is a wrapper
for ODE integrators. As such it can be tested by comparing its results against known analytic
solutions.

### 2 Physical System

For grid functions we integrate

where $n$
is chosen such that a given ODE method can integrate the polynomial exactly, eg.
$n=3$
for the classical Runge-Kutta method. For multi-rate ODE methods, the slow sector integrates

$$\u1e8f=-1+\frac{1}{4}{t}^{m}$$ | (2) |

where $m$ is usually
smaller than $n$,
eg. $2$ for
the RK4-RK2 scheme. Finally grid arrays integrate

$$\u1e8f=-1+\frac{1}{2}{t}^{n}$$ | (3) |