Using Spectral Method as an Approximation for Solving Hyperbolic PDEs

TL;DR

This paper explores the use of spectral methods with Fourier and quantum oscillator bases as an effective numerical approach for solving hyperbolic PDEs, including eigenvalue problems, offering advantages over traditional methods.

Contribution

It introduces spectral methods with specific bases for hyperbolic PDEs, demonstrating their effectiveness and comparing their relative advantages.

Findings

Spectral methods can solve complex hyperbolic PDEs difficult for finite difference methods.

Fourier and quantum oscillator bases each have unique advantages in spectral solutions.

Eigenvalue problems are effectively addressed using this spectral approach.

Abstract

We demonstrate an application of the spectral method as a numerical approximation for solving Hyperbolic PDEs. In this method a finite basis is used for approximating the solutions. In particular, we demonstrate a set of such solutions for cases which would be otherwise almost impossible to solve by the more routine methods such as the Finite Difference Method. Eigenvalue problems are included in the class of PDEs that are solvable by this method. Although any complete orthonormal basis can be used, we discuss two particularly interesting bases: the Fourier basis and the quantum oscillator eigenfunction basis. We compare and discuss the relative advantages of each of these two bases.