## TestMoL

February 12 2014

Abstract

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

 $ẏ=-1+{t}^{n}$ (1)

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

 $ẏ=-1+\frac{1}{4}{t}^{m}$ (2)

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

 $ẏ=-1+\frac{1}{2}{t}^{n}$ (3)