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High order fast numerical methods with linear computational complexity for solving steady state problems of hyperbolic PDEs

Yongtao Zhang
Notre Dame
Website of Yongtao Zhang

Location: Purdue University

Apr 26, 2014 4:00 PM

In this talk, I present our recent studies on developing efficient high order numerical methods for solving steady state problems of two classes of hyperbolic PDEs: static Hamilton-Jacobi equations (especially Eikonal equations) and hyperbolic conservation laws.

The methods we propose have linear computational complexity, namely, the computational cost is O(N) where N is the number of grid points of the computational mesh. For Eikonal equations, we design a third order fast sweeping method with linear computational complexity. This iterative method utilizes the Gauss-Seidel iterations and alternating sweeping strategy to cover a family of characteristics of the Eikonal equations in a certain direction simultaneously in each sweeping order. The method is based on a third order discontinuous Galerkin (DG) finite element solver. Novel causality indicators are designed to guide the information flow directions of the nonlinear Eikonal equations. Numerical experiments show that the method has third order accuracy and a linear computational complexity.

To deal with more general static Hamilton-Jacobi equations, we developed fixed-point fast sweeping methods with high order weighted essentially non-oscillatory (WENO) discretization. This fast sweeping approach can also be applied to solve steady state problems of hyperbolic conservation laws. At last, I shall present a novel method based on the homotopy continuation. A homotopy continuation method is applied to solve polynomial systems resulting from a third order WENO discretization of hyperbolic conservation laws system. Via numerical experiments for both scalar and system problems in one and two dimensions, we show that this new approach has linear computational complexity and is free of the CFL condition constraint. This is joint work with many collaborators.