Abstract
AbstractWe 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.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Numerical Analysis,Analysis
Reference24 articles.
1. Koleva, M.N., Vulkov, L.G.: Numerical solution of the Monge–Ampère equation with an application to fluid dynamics. AIP Conf. Proc. 2048(1), 030002 (2018). https://doi.org/10.1063/1.5082060
2. Caboussat, A.: Computation of portfolio hedging strategies using a reduced Monge-Ampère equation. In: Proceedings of the 20th International Conference on Computing in Economics and Finance, Oslo, June 22–24, 2014, pp. 1–13 (2014)
3. Chen, T.: Smooth local solutions to degenerate hyperbolic Monge–Ampère equations. Ann. PDE 5(1), 1–47 (2019). https://doi.org/10.1007/s40818-018-0055-y
4. Prins, C.R.: Inverse Methods for Illumination Optics. PhD thesis, Eindhoven University of Technology (2014). https://doi.org/10.6100/IR776340
5. Yadav, N.K., Romijn, L.B., ten Thije Boonkkamp, J.H.M., IJzerman, W.L.: A least-squares method for the design of two-reflector optical systems. J. Phys. Photonics 1(3), 034001 (2019). https://doi.org/10.1088/2515-7647/ab2db3
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献