A time-continuous (tc-)embedding method is first proposed for solving nonlinear hyperbolic conservation laws with discontinuous solutions on codimension 1, connected, smooth, and closed one- and two-dimensional manifolds (surface PDEs or SPDEs). The new embedding method improves upon the classical closest point (cp-)embedding method, which requires re-establishments of the constant-along-normal (CAN-)property of the extension function at every Runge-Kutta step, in terms of accuracy and efficiency, by incorporating the CAN-property analytically and explicitly in the embedding equation. The tc-embedding SPDEs are solved by the second-order linear central finite volume scheme with a nonlinear minmod slope limiter and the high-order hybrid central-upwind/WENO finite difference scheme in space, and the third-order total variation diminished Runge-Kutta scheme in time. The hybrid scheme with a troubled-cell detector, based on the radial basis function and Tukey boxplot method, is employed to increase the computational efficiency of solving the 1D SPDE on the 2D manifolds embedded in the 3D Cartesian domain. The solution values at the ghost cells are obtained via an adaptive nonlinear essentially non-oscillatory polynomial interpolation. Numerical results in solving the linear wave equation, the Burgers’ equation, and the Euler equations 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, the Buckley-Leverett equation, classical Riemann problems, and the extended Shu-Osher problems are solved by the hybrid method to demonstrate the robust performance of the tc-embedding method in resolving fine-scale structures efficiently in the presence of a shock and the essentially non-oscillatory capturing of singular structures (shock, contact and rarefaction waves) on simple and complex shaped one- and two-dimensional manifolds.

This is joint work with Dr. Wang Bao-Shan, Li Jiale, Wang Caifeng (Ocean University of China), Prof. Wang Yinghua (Nanjing Tech University), and Prof. Ling Leevan (Hong Kong Baptist University).