Numerical Methods for the Hyperbolic Monge-Amp\`ere Equation Based on the Method of Characteristics

TL;DR

This paper develops numerical methods based on the method of characteristics for solving the hyperbolic Monge-Ampère equation, employing explicit ODE solvers and spline interpolation, with demonstrated numerical performance.

Contribution

It introduces three derivations of the method of characteristics for the Monge-Ampère equation and applies explicit ODE methods for numerical solutions.

Findings

Numerical examples show effective performance of the proposed methods.

Explicit Runge-Kutta and Euler methods are successfully applied.

Spline interpolation enhances the solution accuracy.

Abstract

We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation. The MOC gives rise to two mutually coupled systems of ordinary differential equations. As a special case we consider the Monge-Amp\`ere equation, for which we solve the system of ODE's using explicit one-step methods (Euler, Runge-Kutta) and spline interpolation. Numerical examples demonstrate the performance of the methods.