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.
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.