Differential equations appear in most branches of Physics, and solving
them is an integral part of solving many Physics problems. Although
exact solutions are always better then approximate ones, oftentimes it
is not possible to solve differential equations analytically and
numerical methods must be used. To this end, this web page has been
developed as a tutorial on solving hyperbolic differential equations
numerically using computer tools such as RNPL and DV. The example
chosen to illustrate the numerical techniques is the 3D wave equation
on an arbitrary pseudo-Riemaniann manifold, even though the methods
presented can be adapted for other hyperbolic equations as well.

Mathematical Introduction

The *Mathematical Introduction*
section is aimed at readers who have a background knowledge of
Differential Geometry or General Relativity, and it presents the
mathematical steps used to solve the wave equation, as well as a short
tutorial on coordinate compactification.

Finite Difference Techniques

The *Finite Difference Techniques*
section introduces the formalism required to solve differential
equations numerically, and it should be understandable by anyone who
has a working knowledge of calculus.

Introduction to RNPL

The 3rd section, *Introduction to RNPL*,
explains basic the basic concepts of RNPL, which is a programming
language developed specifically for solving differential equations. It
should be understandable by anyone who has a basic knowledge of
calculus and programming.

Solving the 3D Wave Equation with RNPL

The *3D Wave Equation* section is a continuation of Introduction to RNPL, and it develops more advanced notions. It should be approached after the
1D wave equation example and the Mathematical Introduction have been
studied, as it assumes knowledge of Differential Geometry/General Relativity and of the basic workings of RNPL.

3D Wave Equation Examples | 2D Wave Equation Examples |

Some of the material presented on this web page is based upon work supported in part by the *Princeton Applied and Computational Mathematics Program*
and by the *National Science Foundation* under * Grant No. 0745779 .
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