A Time-Continuous Embedding Method for Scalar Hyperbolic Conservation Laws on Manifolds

Abstract A time-continuous (tc-)embedding method is first proposed for solving nonlinear scalar hyperbolic conservation laws with discontinuous solutions (shocks and rarefaction waves) on codimension 1, connected, smooth, closed manifolds (surface PDEs or SPDEs in $${\mathbb {R}}^2$$ R 2 {R}}^3$$ 3 ). The new embedding improves upon the classical closest point (cp-)embedding method, which requires re-establishments of constant-along-normal (CAN-)property extension function at every time step, terms accuracy efficiency, by incorporating CAN-property analytically explicitly equation. tc-embedding are solved second-order central finite volume scheme a minmod slope limiter space, third-order total variation diminished Runge-Kutta time. An adaptive essentially non-oscillatory polynomial interpolation used to obtain solution values ghost cells. Numerical results linear wave equation Burgers’ show that has better accuracy, improved resolution, reduced CPU times than cp-embedding method. equation, traffic flow problem, Buckley-Leverett demonstrate robust performance resolving fine-scale structures efficiently even presence shock capturing shocks waves simple complex shaped one-dimensional manifolds. also two-dimensional torus-shaped spherical-shaped