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

Author

Listed:
  • 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
    Signify Research)

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.

Suggested Citation

  • M. W. M. C. Bertens & E. M. T. Vugts & M. J. H. Anthonissen & J. H. M. Thije Boonkkamp & W. L. IJzerman, 2022. "Numerical methods for the hyperbolic Monge-Ampère equation based on the method of characteristics," Partial Differential Equations and Applications, Springer, vol. 3(4), pages 1-42, August.
    • Handle: RePEc:spr:pardea:v:3:y:2022:i:4:d:10.1007_s42985-022-00181-4
    DOI: 10.1007/s42985-022-00181-4
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