New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations
Journal of Computational Physics
Weighted ENO Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton--Jacobi equations
Journal of Computational Physics
Adaptive Semidiscrete Central-Upwind Schemes for Nonconvex Hyperbolic Conservation Laws
SIAM Journal on Scientific Computing
Applied Numerical Mathematics - Numerical methods for viscosity solutions and applications
Convex ENO Schemes for Hamilton-Jacobi Equations
Journal of Scientific Computing
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This paper is concerned with computing viscosity solutions of Hamilton---Jacobi equations using high-order Godunov-type projection-evolution methods. These schemes employ piecewise polynomial reconstructions, and it is a well-known fact that the use of more compressive limiters or higher-order polynomial pieces at the reconstruction step typically provides sharper resolution. We have observed, however, that in the case of nonconvex Hamiltonians, such reconstructions may lead to numerical approximations that converge to generalized solutions, different from the viscosity solution. In order to avoid this, we propose a simple adaptive strategy that allows to compute the unique viscosity solution with high resolution. The strategy is not tight to a particular numerical scheme. It is based on the idea that a more dissipative second-order reconstruction should be used near points where the Hamiltonian changes convexity (in order to guarantee convergence to the viscosity solution), while a higher order (more compressive) reconstruction may be used in the rest of the computational domain in order to provide a sharper resolution of the computed solution. We illustrate our adaptive strategy using a Godunov-type central-upwind scheme, the second-order generalized minmod and the fifth-order weighted essentially non-oscillatory (WENO) reconstruction. Our numerical examples demonstrate the robustness, reliability, and non-oscillatory nature of the proposed adaptive method.