Numerical methods for the hyperbolic Monge-Ampère equation based on the method of characteristics

M. W. M. C. Bertens (), E. M. T. Vugts, M. J. H. Anthonissen, J. H. M. Thije Boonkkamp and W. L. IJzerman
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M. W. M. C. Bertens: Eindhoven University of Technology
E. M. T. Vugts: Eindhoven University of Technology
M. J. H. Anthonissen: Eindhoven University of Technology
J. H. M. Thije Boonkkamp: Eindhoven University of Technology
W. L. IJzerman: Eindhoven University of Technology

Partial Differential Equations and Applications, 2022, vol. 3, issue 4, 1-42

Abstract: 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.

Keywords: Hyperbolic PDE; Method of characteristics; Monge–Ampère; One-step methods; Boundary conditions; 35L70; 65M25; 65N22; 34B15 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-022-00181-4

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