Positivity-Preserving Flux Difference Splitting Schemes

International journal for …, 2001

New results are presented here for finite volume (FV) methods that use flux vector splitting (FVS) along with higher-order reconstruction schemes. Apart from spectral accuracy of the resultant methods, the numerical stability is investigated which restricts the allowable time step or the Courant -Friedrich -Lewy (CFL) number. Also the dispersion relation preservation (DRP) property of various spatial and temporal discretization schemes is investigated. The DRP property simultaneously fixes space and time steps. This aspect of numerical schemes is important for simulation of high-Reynolds number flows, compressible flows with shock(s) and computational aero-acoustics. It is shown here that for direct numerical simulation applications, the DRP property is more restrictive than stability criteria.

Journal of Physics: Conference Series, 2019

In this work, we have carried out the assessment of a high resolution scheme for unsteady compressible flow. For high order spatial accuracy, we have used fifth order weighted essentially non oscillatory (WENO) scheme. This scheme is applied to four flux difference splitting (FDS) methods: Harten-Lax-van-Leer (HLL), Roe solver, Harten-Lax-van Leer-Contact (HLLC), and Rusanov methods. We have compared results of these flux schemes with each other. WENO scheme is used for the reconstruction of left and right state variable across the cell interface for high resolution. The reconstruction procedure is performed in terms of primitive variables instead of conservative variable, in order to avoid spurious oscillation. We have considered two test cases: shock wave reflection and supersonic viscous flow over a flat plate, to access the performance of FDS schemes. An explicit third order TVD Runge-Kutta method is used for advancement of solution in time. The present results are compared with available numerical solutions. WENO-HLLC has good shock capturing capabilities as compare to WENO-Roe, WENO-HLL and WENO-Rusanov methods. It also provides best results inside and outside the boundary layer.

Journal of Computational Physics, 2005

A new high resolution finite volume method is proposed here that uses flux-vector splitting as the building block to represent the physical processes. The high accuracy projection in space is obtained here, inter alia, by using some compact schemes of Sengupta et al. and Lele [2] -that have extremely high accuracy in spectral plane. This produces a new class of finite volume (FV) scheme for spatial discretization, that has been analyzed for spectral accuracy, numerical stability and dispersion relation preservation (DRP) property following the full-domain analysis method of [1] using two different time integration strategies. These combination of spatial and temporal methods have been tested for two linear wave problems, reported in [1,3] and the solution of BurgersÕ equation is compared with the analytical results presented in Adams and Shariff . The shock capturing method is based on the proposed technique in , that does not require the usage of any nonlinear differencing techniques or limiters.

Journal of Computational Physics, 2003

The aim of this paper is to construct hybrid flux vector splitting (FVS) and flux difference splitting (FDS) schemes for a commonly used two-fluid model consisting of two separate momentum equations. This is done by refining ideas previously applied to develop hybrid FVS/FDS schemes for a simpler two-phase model consisting of a mixture momentum equation [J. Comput. Phys. 175 (2002) 674]. More specifically, we seek to construct upwind type of schemes which are not based on calculations of the full eigenstructure of Jacobi matrices as needed by approximate Riemann solvers like the Roe scheme. Based on a crude approximation of the eigenstructure of the model, we derive schemes of the van Leer and FVS type. We demonstrate that these schemes possess desirable stability properties, but are excessively diffusive. By adapting ideas originally suggested by Wada and Liou [SIAM J. Sci. Comput. 18 (1997) 633] for the Euler equations, we suggest a mechanism for removing numerical dissipation. We present numerical simulations where we compare the performance of the resulting schemes with that of the Roe scheme, and by that shed light on the issues of accuracy, efficiency, and robustness of the proposed schemes. Particularly, we consider the classical water faucet problem as well as a stiff separation problem which locally involves transition from two-phase to single-phase flow. Results from these test cases show that we are able to construct hybrid FVS/FDS schemes which properly combine the accuracy of FDS in the resolution of sharp mass fronts and the robustness of FVS which ensures stability under stiff conditions.

This paper presents a new flux splitting scheme for the Euler equations. The proposed scheme termed TV-HLL is obtained by following the Toro-Vazquez splitting (Toro and Vázquez-Cendón, 2012) and using the HLL algorithm with modified wave speeds for the pressure system. Here, the intercell velocity for the advection system is taken as the arithmetic mean. The resulting scheme is more accurate when compared to the Toro-Vazquez schemes and also enjoys the property of recognition of contact discontinuities and shear waves. Accuracy, efficiency, and other essential features of the proposed scheme are evaluated by analyzing shock propagation behaviours for both the steady and unsteady compressible flows. The accuracy of the scheme is shown in 1D test cases designed by Toro.

Journal of Computational Physics

A new class of flux-limited schemes for systems of conservation laws is presented that is both high-resolution and positivity-preserving. The schemes are obtained by extending the Steger-Warming method to second-order accuracy through the use of component-wise TVD flux limiters while ensuring that the coefficients of the discretization equation are positive. A coefficient is considered positive if it has all-positive eigenvalues and has the same eigenvectors as those of the convective flux Jacobian evaluated at the corresponding node. For certain systems of conservation laws, such as the Euler equations for instance, this condition is sufficient to guarantee positivity-preservation. The method proposed is advantaged over previous positivity-preserving flux-limited schemes by being capable to capture with high resolution all wave types (including contact discontinuities, shocks, and expansion fans). Several test cases are considered in which the Euler equations in generalized curvilinear coordinates are solved in 1D, 2D, and 3D. The test cases confirm that the proposed schemes are positivity-preserving while not being significantly more dissipative than the conventional TVD methods. The schemes are written in general matrix form and can be used to solve other systems of conservation laws, as long as they are homogeneous of degree one.

Healing of nonphysical flow solutions and shock instability from the use of Roe's flux-difference splitting scheme is presented. The proposed method heals nonphysical flow solutions such as the carbuncle phenomenon, the shock instability from the odd-even decoupling problem, and the expansion shock generated from the violated entropy condition. The performance and efficiency of the proposed method are evaluated by solving several benchmark and complex high-speed compressible flow problems.

International Journal for Numerical Methods in Fluids, 1988

The flux-vector splitting method is applied to the convective part of the steady Navier-Stokes equations for incompressible flow. By the use of partial upwind differences in the split first-order part and central differences in the second-order part, a set of discrete equations is obtained which can be solved by vector variants of classical relaxation schemes. It is shown that accurate results can be obtained on one of the GAMM backward-facing step test problems.

The extension of a flux discretization method to second-order accuracy can lead to some difficulties in maintaining positivity preservation. While the MUSCL-TVD scheme maintains the positivity preservation property of the underlying 1st-order flux discretization method, a flux-limited-TVD scheme does not. A modification is here proposed to the flux-limited-TVD scheme to make it positivity-preserving when used in conjunction with the Steger-Warming flux vector splitting method. The proposed algorithm is then compared to MUSCL for several test cases. Results obtained indicate that while the proposed scheme is more dissipative in the vicinity of contact discontinuities, it performs significantly better than MUSCL when solving strong shocks in hypersonic flowfields: the amount of pressure overshoot downstream of the shock is minimized and the time step can be set to a value typically two or three times higher. While only test cases solving the one-dimensional Euler equations are here presented, the proposed scheme is written in general form and can be extended to other physical models.

1992

We introduce in this paper a new concept for upwinding: the Hybrid Upwind Splitting (HUS). This original strategy for upwinding is achieved by combining the two previous existing approaches, the Flux Vector (FVS) and Flux Difference Splittings (FDS), while retaining their own interesting features. Indeed, our approach yields upwind methods that share the robustness of FVS schemes in the capture of nonlinear waves and the accuracy of some FDS schemes in the capture of linear waves. We describe here some examples of such HUS methods obtained by hybridizing the Osher approach with FVS schemes. Numerical illustrations are displayed and will prove in particular the relevance of the HUS methods we propose for viscous calculations.