A time-continuous embedding method for scalar hyperbolic conservation laws on manifolds
A new closest point TC (Time-Continuous) embedding method is developed for solving nonlinear scalar hyperbolic conservation law (PDEs) on one-dimensional, connected, smooth, and closed manifolds (SPDEs). It improves upon the classical CP (Closest Point) embedding method that requires a costly re-establishment of the extension function's CAN (Constant-Along-Normal) property at every time step; the TC embedding method incorporates the CAN-property analytically and explicitly in the SPDEs directly. The TC embedding SPDEs are solved by the second-order nonlinear central finite volume scheme with a nonlinear minmod slope limiter in space, and the third-order total variation diminished Runge-Kutta scheme in time. An adaptive nonlinear essentially non-oscillatory polynomial interpolation is used to obtain the solution values at the ghost cells. Numerical results in solving the linear wave equation and the Burgers' equation show that the proposed TC embedding method has better accuracy, improved resolution, and reduced CPU times than the classical CP embedding method. The Burgers' equation, the traffic flow problem, and the Buckley-Leverett equation are solved to demonstrate the efficiency of the TC embedding method in resolving fine-scale structures efficiently even in the presence of a shock and the essentially non-oscillatory capturing of shocks and rarefaction waves on simple and complex shaped one-dimensional manifolds. Burgers' equation is also solved on the two-dimensional torus-shaped and spherical-shaped manifolds.
scientific article; zbMATH DE number 7632683
scientific article