Central Runge--Kutta Schemes for Conservation Laws

Journal of Computational Physics, 1998

This is the fifth paper in a series in which we construct and study the so-called Runge-Kutta Discontinuous Galerkin method for numerically solving hyperbolic conservation laws. In this paper, we extend the method to multidimensional nonlinear systems of conservation laws. The algorithms are described and discussed, including algorithm formulation and practical implementation issues such as the numerical fluxes, quadrature rules, degrees of freedom, and the slope limiters, both in the triangular and the rectangular element cases. Numerical experiments for two dimensional Euler equations of compressible gas dynamics are presented that show the effect of the (formal) order of accuracy and the use of triangles or rectangles, on the quality of the approximation.

Journal of Computational Physics, 2015

In this paper, an upwind space-time conservation element and solution element (CE/SE) method is developed to solve conservation laws. In the present method, the mesh quantity and spatial derivatives are the independent marching variables, which is consistent with the original CE/SE method proposed by Chang (1995) [5]. The staggered time marching strategy and the definition of conservation element (CE) also follow Chang's propositions. Nevertheless, the definition of solution element (SE) is modified from that of Chang. The numerical flux through the interface of two different conservation elements is not directly derived by a Taylor expansion in the reversed time direction as proposed by Chang, but determined by an upwind procedure. This modification does not change the local and global conservative features of the original method. Although, the time marching scheme of mesh variables is the same with the original method, the upwind fluxes are involved in the calculation of spatial derivatives, yielding a totally different approach from that of Chang's method. The upwind procedure breaks the space-time inversion invariance of the original scheme, so that the new scheme can be directly applied to capture discontinuities without spurious oscillations. In addition, the present method maintains low dissipation in a wide range of CFL number (from 10 −6 to 1). Furthermore, we extend the upwind CE/SE method to solve the Euler equations by adopting three different approximate Riemann solvers including Harten, Lax and van Leer (HLL) Riemann solver, contact discontinuity restoring HLLC Riemann solver and mathematically rigorous Roe Riemann solver. Extensive numerical examples are carried out to demonstrate the robustness of the present method. The numerical results show that the new CE/SE solvers perform improved resolutions.

World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 2011

The ultimate goal of this article is to develop a robust and accurate numerical method for solving hyperbolic conservation laws in one and two dimensions. A hybrid numerical method, coupling a cheap fourth order total variation diminishing (TVD) scheme [1] for smooth region and a Robust seventh-order weighted non-oscillatory (WENO) scheme [2] near discontinuities, is considered. High order multi-resolution analysis is used to detect the high gradients regions of the numerical solution in order to capture the shocks with the WENO scheme, while the smooth regions are computed with fourth order total variation diminishing (TVD). For time integration, we use the third order TVD Runge-Kutta scheme. The accuracy of the resulting hybrid high order scheme is comparable with these of WENO, but with significant decrease of the CPU cost. Numerical demonstrates that the proposed scheme is comparable to the high order WENO scheme and superior to the fourth order TVD scheme. Our scheme has the added advantage of simplicity and computational efficiency. Numerical tests are presented which show the robustness and effectiveness of the proposed scheme.

Numerical Methods for Partial Differential Equations, 2005

We present a family of central-upwind schemes on general triangular grids for solving two-dimensional systems of conservation laws. The new schemes enjoy the main advantages of the Godunov-type central schemes-simplicity, universality, and robustness and can be applied to problems with complicated geometries. The "triangular" central-upwind schemes are based on the use of the directional local speeds of propagation and are a generalization of the central-upwind schemes on rectangular grids, recently introduced in Kurganov et al. [SIAM J Sci Comput 23 (2001), 707-740]. We test a second-order version of the proposed scheme on various examples. The main purpose of the numerical experiments is to demonstrate the potential of our method. The more universal "triangular" central-upwind schemes provide the same high accuracy and resolution as the original, "rectangular" ones, and at the same time, they can be used to solve hyperbolic systems of conservation laws on complicated domains, where the implementation of triangular or mixed grids is advantageous.

SIAM Journal on Scientific Computing, 2003

This paper considers a family of nonconservative numerical discretizations for conservation laws which retain the correct weak solution behavior in the limit of mesh refinement whenever sufficient-order numerical quadrature is used. Our analysis of 2-D discretizations in nonconservative form follows the 1-D analysis of Hou and Le Floch [Math. Comp., 62 (1994), pp. 497-530]. For a specific family of nonconservative discretizations, it is shown under mild assumptions that the error arising from nonconservation is strictly smaller than the discretization error in the scheme. In the limit of mesh refinement under the same assumptions, solutions are shown to satisfy a global entropy inequality. Using results from this analysis, a variant of the "N" (Narrow) residual distribution scheme of van der Weide and Deconinck [Computational Fluid Dynamics '96, Wiley, New York, 1996, pp. 747-753] is developed for first-order systems of conservation laws. The modified form of the Nscheme supplants the usual exact single-state mean-value linearization of flux divergence, typically used for the Euler equations of gasdynamics, by an equivalent integral form on simplex interiors. This integral form is then numerically approximated using an adaptive quadrature procedure. This quadrature renders the scheme nonconservative in the sense described earlier so that correct weak solutions are still obtained in the limit of mesh refinement. Consequently, we then show that the modified form of the N-scheme can be easily applied to general (nonsimplicial) element shapes and general systems of first-order conservation laws equipped with an entropy inequality, where exact mean-value linearization of the flux divergence is not readily obtained, e.g., magnetohydrodynamics, the Euler equations with certain forms of chemistry, etc. Numerical examples of subsonic, transonic, and supersonic flows containing discontinuities together with multilevel mesh refinement are provided to verify the analysis.

2008

We propose a new fourth-order non-oscillatory central scheme for computing approximate solutions of hyperbolic conservation laws. A piecewise cubic polynomial is used for the spatial reconstruction and for the numerical derivatives we choose genuinely fourth-order accurate non-oscillatory approximations. The solution is advanced in time using natural continuous extension of Runge-Kutta methods. Numerical tests on both scalar and gas dynamics problems confirm that the new scheme is non-oscillatory and sharper than existing fourth-order central schemes when solving profiles with discontinuities. Experiments on nonlinear Burgers' equation indicate that our scheme is superior to existing fourth-order central schemes in the sense that the total variation of the computed solutions are closer to the total variation of the exact solution.

Journal of Computational and Applied Mathematics - J COMPUT APPL MATH, 2007

In this article we present a high resolution hybrid central finite difference—WENO scheme for the solution of conservation laws, in particular, those related to shock–turbulence interaction problems. A sixth order central finite difference scheme is conjugated with a fifth order weighted essentially non-oscillatory WENO scheme in a grid-based adaptive way. High order multi-resolution analysis is used to detect the high gradients regions of the numerical solution in order to capture the shocks with the WENO scheme while the smooth regions are computed with the more efficient and accurate central finite difference scheme. The application of high order filtering to mitigate the dispersion error of central finite difference schemes is also discussed. Numerical experiments with the 1D compressible Euler equations are shown.

Siam Journal on Scientific Computing, 2000

We present a new third-order central scheme for approximating solutions of systems of conservation laws in one and two space dimensions. In the spirit of Godunov-type schemes, our method is based on reconstructing a piecewisepolynomial interpolant from cell-averages which is then advanced exactly in time.

Communications in Applied Mathematics and Computational Science, 2011

We present a fourth-order accurate finite-volume method for solving time-dependent hyperbolic systems of conservation laws on Cartesian grids with multiple levels of refinement. The underlying method is a generalization of that developed by Colella, Dorr, Hittinger and Martin (2009) to nonlinear systems, and is based on using fourth-order accurate quadratures for computing fluxes on faces, combined with fourth-order accurate Runge-Kutta discretization in time. To interpolate boundary conditions at refinement boundaries, we interpolate in time in a manner consistent with the individual stages of the Runge-Kutta method, and interpolate in space by solving a least-squares problem over a neighborhood of each target cell for the coefficients of a cubic polynomial. The method also uses a variation on the extremum-preserving limiter of Colella and Sekora (2008), as well as slope flattening and a fourth-order accurate artificial viscosity for strong shocks. We show that the resulting method is fourth-order accurate for smooth solutions, and is robust in the presence of complex combinations of shocks and smooth flows. 1. High-order finite-volume methods In the finite-volume approach, the spatial domain in ‫ޒ‬ D is discretized as a union of rectangular control volumes that covers the spatial domain. For Cartesian-grid finite-volume methods, a control volume V i takes the form V i = [i h, (i + u)h] for i ∈ ‫ޚ‬ D , u = (1, 1,. .. , 1), where h is the grid spacing. A finite-volume discretization of a partial differential equation is based on averaging that equation over control volumes, applying the divergence theorem to replace volume integrals by integrals over the boundary of the control volume,

SIAM Journal on Scientific Computing, 2000

We present a new third-order, semi-discrete, central method for approximating solutions to multi-dimensional systems of hyperbolic conservation laws, convectiondiffusion equations, and related problems. Our method is a high-order extension of the recently proposed second-order, semi-discrete method in .