Numerical Solution of an Extra-wide Angle Parabolic Equation through Diagonalization of a 1-D Indefinite Schr\"{o}dinger Operator with a Piecewise Constant Potential

TL;DR

This paper introduces a numerical method for solving an extra-wide angle parabolic PDE by diagonalizing a 1-D indefinite Schrödinger operator with a piecewise constant potential, enabling efficient eigenfunction expansion computation.

Contribution

The work develops a scalable eigenvalue estimation approach for indefinite Schrödinger operators with piecewise potentials, improving computational efficiency for solving related PDEs.

Findings

Eigenvalue estimates enable efficient eigenfunction expansion.

Method's accuracy and scalability demonstrated through numerical experiments.

Computational cost per eigenpair is independent of grid size.

Abstract

We present a numerical method for computing the solution of a partial differential equation (PDE) for modeling acoustic pressure, known as an extra-wide angle parabolic equation, that features the square root of a differential operator. The differential operator is the negative of an indefinite Schr\"{o}dinger operator with a piecewise constant potential. This work primarily deals with the 3-piece case; however, a generalization is made the case of an arbitrary number of pieces. Through restriction to a judiciously chosen lower-dimensional subspace, approximate eigenfunctions are used to obtain estimates for the eigenvalues of the operator. Then, the estimated eigenvalues are used as initial guesses for the Secant Method to find the exact eigenvalues, up to roundoff error. An eigenfunction expansion of the solution is then constructed. The computational expense of obtaining each eigenpair is independent of the grid size. The accuracy, efficiency, and scalability of this method is shown through numerical experiments and comparisons with other methods.