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Numerical approximation techniques for a fully nonlinear Schrödinger equation: nonlinear waves and shocks
Uhl, Townes Strong, Scott A.
Uhl, Townes
Strong, Scott A.
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2025-04
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In this project, we study three numerical methods: the Finite Difference (FD) Method, a Finite Volume (FV) Scheme with an Osher-Solomon Riemann Problem Approximator, and the Entropy Viscosity Method. These methods are applied to a fully nonlinear partial differential equation (PDE) of Schrödinger type, which is suspected to produce shocks in finite time. Our goal is to identify which terms in the PDE drive shock formation and to develop a stability-preserving scheme for approximating solutions to the resulting balance law. The theory of stability-preserving methods, e.g., entropy-viscosity method, is primarily developed for hyperbolic conservation laws. Therefore, we also explore potential reformulations of the model PDE to recast it as a hyperbolic system of PDEs. Finally, we validate our methods by solving Burgers' equation, which has a known analytical solution.
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