Finite volume schemes for Hamilton-Jacobi equations

Abstract

We introduce two classes of monotone finite volume schemes for Hamilton-Jacobi equations. The corresponding approximating functions are piecewise linear defined on a mesh consisting of triangles. The schemes are shown to converge to the viscosity solution of the Hamilton-Jacobi equation. Mathematics Subject Classification (1991): 65M06, 65M12 IntroductionIn this paper we consider finite volume schemes approximating the viscosity solution of the Hamilton-Jacobi equationwhere the Hamiltonian H ∈ C 0,1 (R N ), the space of Lipschitz continuous functions in R N , and u 0 ∈ BU C(R N ), the space of bounded and uniformly continuous functions on R N . A few comments about viscosity solutions are in order here. The class of viscosity solutions, which was introduced by Crandall and Lions in [CL1], is the "correct class" of weak solutions for Hamilton-Jacobi equations as well as fully nonlinear possibly degenerate second order elliptic and parabolic pde. We refer to the book of Barles [B] and the "User's Guide" of Crandall,

A finite volume method based on the Crouzeix-Raviart element for elliptic PDE's in two dimensions

“…Indeed in §4 we consider a finite volume method for the approximation of the solution u of problem (1.1) and in Theorems 4.1 and 4.2 we extend our convergence results to this case. In [13], Kossioris, Makridakis and Souganidis use a similar method to construct finite volume approximations to Hamilton-Jacobi equations.…”

Central WENO Schemes for Hamilton–Jacobi Equations on Triangular Meshes

“…These include the ENO schemes of Osher, Sethian, and Shu [30,31] and the WENO schemes of Jiang and Peng [16]. Similar methods on triangular meshes include the pioneering works of Abgrall [1,2], Augoula and Abgrall [3], Barth and Sethian [5], Kossioris, Makridakis, and Souganidis [19], and the recent work of Zhang and Shu [35] on WENO schemes for HJ equations on triangular meshes. The high-order WENO reconstructions on triangular meshes that were used in [35] were based on the results of Hu and Shu [15].…”

Semi-discrete Schemes for Hamilton-Jacobi Equations on Unstructured Grids

“…The numerical fluxes of Abgrall were combined with WENO reconstructions on triangular meshes by Hu and Shu in [20]. Another finite-volume scheme on unstructured grids was proposed by Kossioris et al in [9].…”