Numerical methods for the hyperbolic Monge-Ampère 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 (PDE) in two independent variables. The MOC gives rise to two mutually coupled systems of ordinary differential equations (ODEs). As a special case we...

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Bibliographic Details
Published in:SN partial differential equations and applications 2022-08, Vol.3 (4), Article 52
Main Authors: Bertens, M. W. M. C., Vugts, E. M. T., Anthonissen, M. J. H., ten Thije Boonkkamp, J. H. M., IJzerman, W. L.
Format: Article
Language:English
Subjects:
2. Computational Approaches
Analysis
Mathematical and Computational Biology
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Original Paper
Partial Differential Equations
Theoretical
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Summary: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.
ISSN:2662-2963
2662-2971
DOI:10.1007/s42985-022-00181-4