Numerical Methods for Solving Hyperbolic Wave Equations (by B. Stoica)
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.
The Examples present graphical solutions to the wave equation in 2 and 3 dimensions, obtained with the techniques described in this web page. No mathematical or programming background is required for an intuitive understanding of the solutions.
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.