Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems

Stochastic physical problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zero of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods

Riemann solvers in relativistic astrophysics

AbstractOur contribution reviews High Resolution Shock Capturing methods (HRSC) in the field of relativistic hydrodynamics with special emphasis on Riemann solvers. HRSC techniques achieve highly accurate numerical approximations (formally second order or better) in smooth regions of the flow, and capture the motion of unresolved steep gradients without creating spurious oscillations. One objective of our contribution is to show how these techniques have been extended to relativistic hydrodynamics, making it possible to explore some challenging astrophysical scenarios. We will review recent literature concerning the main properties of different special relativistic Riemann solvers, and discuss several 1D and 2D test problems which are commonly used to evaluate the performance of numerical methods in relativistic hydrodynamics. Finally, we will illustrate the use of HRSC methods in several applications in special and general relativistic hydrodynamics

Relativistic MHD and black hole excision: Formulation and initial tests

A new algorithm for solving the general relativistic MHD equations is described in this paper. We design our scheme to incorporate black hole excision with smooth boundaries, and to simplify solving the combined Einstein and MHD equations with AMR. The fluid equations are solved using a finite difference Convex ENO method. Excision is implemented using overlapping grids. Elliptic and hyperbolic divergence cleaning techniques allow for maximum flexibility in choosing coordinate systems, and we compare both methods for a standard problem. Numerical results of standard test problems are presented in two-dimensional flat space using excision, overlapping grids, and elliptic and hyperbolic divergence cleaning.Comment: 22 pages, 8 figure

The Explicit Simplified Interface Method for compressible multicomponent flows

This paper concerns the numerical approximation of the Euler equations for multicomponent flows. A numerical method is proposed to reduce spurious oscillations that classically occur around material interfaces. It is based on the "Explicit Simplified Interface Method" (ESIM), previously developed in the linear case of acoustics with stationary interfaces (2001, J. Comput. Phys. 168, pp.~227-248). This technique amounts to a higher order extension of the "Ghost Fluid Method" introduced in Euler multicomponent flows (1999, J. Comput. Phys. 152, pp. 457-492). The ESIM is coupled to sophisticated shock-capturing schemes for time-marching, and to level-sets for tracking material interfaces. Jump conditions satisfied by the exact solution and by its spatial derivative are incorporated in numerical schemes, ensuring a subcell resolution of material interfaces inside the meshing. Numerical experiments show the efficiency of the method for rich-structured flows.Comment: to be published in SIAM Journal of Scientific Computing (2005