Lax–Friedrichs fast sweeping methods for steady state problems for hyperbolic conservation laws
Introduction
Hyperbolic conservation laws and Hamilton–Jacobi equations are first order nonlinear partial differential equations which arise in many applications such as gas dynamics [19], [40], shallow water waves [41], geometrical optics [42], crystal growth [23], [25], etching, photolithography [33], computer vision [32], [24] and seismology [7]. The solutions of these equations can develop singularity, such as discontinuity in the solutions or their derivatives. Under these circumstances, the solutions do not satisfy the equations in the classical sense and one must resort to weak solutions. For hyperbolic conservation equations, “vanishing viscosity solution” and “entropy solution” are introduced to define the weak solution uniquely (see [20], [18] and references therein). In early 1980s, Crandall and Lions [10], [11] introduced “viscosity solution”, among weak solutions, to study of the existence, uniqueness, and stability properties of Hamilton–Jacobi equations. Since then, many numerical methods have been proposed to approximate the viscosity solutions [34], [18], [9], [38]. The challenges of the numerical schemes generally are to achieve high order accuracy and be able to resolve the shock or singularity well, while maintaining conservation. One key to the scheme design of hyperbolic conservation laws and Hamilton–Jacobi equations, in order to correctly capture the viscosity solution, is the use of consistent and conservative numerical fluxes and Hamiltonians.
For Hamilton–Jacobi equations,after Osher proved in [22] the link between static and time dependent Hamilton–Jacobi equations, in which the zero level set of the viscosity solution of the time dependent problem at a later time t is the t-level set of the static problem, fast sweeping methods for static Hamilton–Jacobi equations became popular due to their high efficiency. A fast sweeping method mainly consists of the following three essential ingredients: (1) an efficient local solver on a given Cartesian mesh [39], [46], [15], [17], [45], [26] or triangulation [26], [16] based on monotone numerical Hamiltonians, (2) systematic orderings of solution nodes according to some pre-determined information-flowing directions, and (3) Gauss–Seidel type iterations based on a given order of solution nodes. Among fast sweeping schemes, the methods based on upwind Hamiltonians are most efficient for convex Hamiltonians [39], [46], while the methods based on Lax–Friedrichs fluxes [15], [45] are most flexible to deal with general non-convex Hamiltonians.
Although for hyperbolic conservation laws with source terms,in which the Jacobian matrix is diagonalizable with all the eigenvalues being real for any u, there is no such link between time dependent problems and static problems as in Hamilton–Jacobi equations, it is still favorable to have efficient numerical methods for steady state hyperbolic problems. A class of schemes, called “residual distribution schemes” [1], [2], [3], [12], [29], [37], were proposed for solving steady state problems with pseudo time stepping. The spirit of these methods is to distribute the residuals, defined through integrating the flux and source terms on triangular or quadrilinear cells in a conservative fashion, and march in pseudo time. Residual distribution schemes were later generalized to high order schemes by Abgrall and Roe [4] through a finite element based approach. Based on the same distribution principles, Chou and Shu [8] developed a finite difference based residual distribution scheme which works on curvilinear meshes, and their scheme achieves high order accuracy and low computational cost as in finite difference methods, but without the constraint of uniform meshes.
These residual distribution schemes, though more efficient by using only pseudo time stepping, are still greatly constrained by the CFL condition for stability, which can substantially limit the speed of the schemes. In this paper, we develop a Gauss–Seidel type iterative method to accelerate the speed to compute the steady state solutions of hyperbolic equations. Inspired by fast sweeping methods for time independent Hamilton–Jacobi equations, we propose methods which discretize the steady state hyperbolic conservation laws directly, by approximating the spatial derivatives with consistent and conservative numerical fluxes, and iterate with Gauss–Seidel type nonlinear method with a finite number of alternating sweeping directions. In particular, we use the Lax–Friedrichs fluxes evaluated in WENO (Weighted Essentially Non-oscillatory) fashion [35], [36], [21], [14], [34], to achieve high order accuracy as well as high resolution of shocks.
It is worth pointing out here that, while most steady state hyperbolic problems have unique steady states, with initial conditions reasonably perturbed from the solutions, there are some problems whose steady states are totally dependent on the initial conditions through mass conservation [30]. In those cases, Gauss–Seidel type sweeping may not conserve the mass through the iterations, and therefore an additional constraint needs to be imposed in order to select the particular steady state.
This paper is organized as follows. In Section 2, we present the high order Lax–Friedrichs sweeping methods for one-dimensional scalar and system problems. In Sections 3, the method is extended to two-dimensional equations. Section 4 describes an efficient accuracy-preserving stopping criterion for the fast sweeping iterative scheme. Section 5 contains extensive numerical simulations for one and two-dimensional scalar and system steady state problems to demonstrate high order accuracy, efficiency and robustness of our scheme. Conclusions are given in Section 6.
Section snippets
One-dimensional scalar problems
In this section, we consider the one-dimensional scalar steady state problemsubject to an initial guess and appropriate boundary conditions.
To obtain a numerical scheme, the interval is first discretized uniformly into N cells, and the grid points are denoted by , where and . The midpoint of a cell is defined as , . The numerical approximation of u on the grid points are denoted by . A conservative
Two-dimensional problems
The sweeping method we described in the previous sections can be easily extended to the two-dimensional steady state problemLet denote the grid points of a uniform discretization of the computational domain, with and as the mesh sizes for x and y direction, respectively. We use to represent the numerical solution of u at grid point . A conservative finite difference discretization of (14) can
Stopping criteria for convergence
Iterative schemes always require stopping criteria by which the convergence of the numerical scheme is determined. Traditional Gauss–Seidel iterations adopt the stopping criteria in which the algorithm stops when a particular norm of the difference of successive iterations, called residue, is smaller than a number. This stopping criteria is effective when the residue of the scheme is monotonically decreasing, but not robust enough for schemes with oscillatory convergence history, such as high
Numerical results
Here we show the numerical results of the proposed Lax–Friedrichs WENO sweeping method for hyperbolic conservation laws with source terms in both scalar and system test problems in one and two dimensions. The efficiency and high order accuracy of the proposed scheme will be demonstrated.
All the spatial discretizations in our numerical results are uniform. The values of in Lax–Friedrichs fluxes and in the flux splitting are updated at the beginning of each directional sweeping and relaxed by
Conclusion
In this paper, we proposed the first fast sweeping method for solving steady state hyperbolic equations with source terms. The method is based on the Lax–Friedrichs fluxes with WENO (Weighted Essentially Non-oscillatory) reconstruction to achieve high order accuracy. The alternating sweeping directions with Gauss–Seidel iterative updates are used to accelerate the speed of convergence. Furthermore, the modified stopping criteria is proposed to stop the algorithm when the residue of the scheme
Acknowledgement
We thank reviewers for their constructive comments and Professor Chi-Wang Shu for the valuable discussion.
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