Abstract
We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation (PDE) in two independent variables. The MOC gives rise to two mutually coupled systems of ordinary differential equations (ODEs). As a special case we consider the Monge–Ampère (MA) equation, for which we present a general method of determining the location and number of required boundary conditions. We solve the systems of ODEs using explicit one-step methods (Euler, Runge-Kutta) and spline interpolation. Reformulation of the Monge–Ampère equation as an integral equation yields via its residual a proxy for the error of the numerical solution. Numerical examples demonstrate the performance and convergence of the methods.
| Original language | English |
|---|---|
| Article number | 52 |
| Number of pages | 42 |
| Journal | Partial Differential Equations and Applications |
| Volume | 3 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 11 Jul 2022 |
Funding
This work is part of the research programme Nederlandse Organisatie voor Wetenschappelijk Onderzoek Toegepaste en Technische Wetenschappen (NWO-TTW) Perspectief with project number P15-36, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO). Special thanks goes to J. de Graaf, for contributing Appendix B, which has greatly simplified generating test examples for the Monge–Ampère equation. Contact: CASA, Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, The Netherlands. This work is part of the research programme Nederlandse Organisatie voor Wetenschappelijk Onderzoek Toegepaste en Technische Wetenschappen (NWO-TTW) Perspectief with project number P15-36, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).
Keywords
- Boundary conditions
- Hyperbolic PDE
- Method of characteristics
- Monge–Ampère
- One-step methods