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

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.