Multiscale Central High-Resolution Schemes with Different Types of Extended Slope Limiters on Wavelet-Based Adapted Cells

High-resolution finite volume methods for solving systems of conservation laws have been widely embraced in research areas ranging from astrophysics to geophysics and aero-thermodynamics. These methods are typically at least second-order accurate in space and time, deliver non-oscillatory solutions in the presence of near discontinuities, e.g., shocks, and introduce minimal dispersive and diffusive effects. High-resolution methods promise to provide greatly enhanced solution methods for Sandia's mainstream shock hydrodynamics and compressible flow applications, and they admit the possibility of a generalized framework for treating multi-physics problems such as the coupled hydrodynamics, electro-magnetics and radiative transport found in Z pinch physics. In this work, we describe initial efforts to develop a generalized 'black-box' conservation law framework based on modern high-resolution methods and implemented in an object-oriented software framework. The framework is based on the solution of systems of general non-linear hyperbolic conservation laws using Godunov-type central schemes. In our initial efforts, we have focused on central or central-upwind schemes that can be implemented with only a knowledge of the physical flux function and the minimal/maximal eigenvalues of the Jacobian of the flux functions, i.e., they do not rely on extensive Riemann decompositions. Initial experimentation with high-resolution central schemes suggests that contact discontinuities with the concomitant linearly degenerate eigenvalues of the flux Jacobian do not pose algorithmic difficulties. However, central schemes can produce significant smearing of contact discontinuities and excessive dissipation for rotational flows. Comparisons between 'black-box' central schemes and the piecewise parabolic method (PPM), which relies heavily on a Riemann decomposition, shows that roughly equivalent accuracy can be achieved for the same computational cost with both methods. However, PPM clearly outperforms the central schemes in terms of accuracy at a given grid resolution and the cost of additional complexity in the numerical flux functions. Overall we have observed that the finite volume schemes, implemented within a well-designed framework, are extremely efficient with (potentially) very low memory storage. Finally, we have found by computational experiment that second and third-order strong-stability preserving (SSP) time integration methods with the number of stages greater than the order provide a useful enhanced stability region. However, we observe that non-SSP and non-optimal SSP schemes with SSP factors less than one can still be very useful if used with time-steps below the standard CFL limit. The 'well-designed' integration schemes that we have examined appear to perform well in all instances where the time step is maintained below the standard physical CFL limit.

Multiresolution-based adaptive central high resolution schemes for modeling of nonlinear propagating fronts

High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method

New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection–Diffusion Equations

In this study, average-interpolating radial basis functions (RBFs) are successfully integrated with central high-resolution schemes to achieve a higher-order central method. This proposed method is used for simulation of generalized coupled thermoelasticity problems including shock (singular) waves in their solutions. The thermoelasticity problems include the LS (systems with one relaxation parameter) and GN (systems without energy dissipation) theories with constant and variable coefficients. In the central high resolution formulation, RBFs lead to a reconstruction with the optimum recovery with minimized roughness on each cell: this is essential for oscillation-free reconstructions. To guarantee monotonic reconstructions at cell-edges, the nonlinear scaling limiters are used. Such reconstructions, finally, lead to the total variation bounded (TVB) feature. As RBFs work satisfactory on non-uniform cells/grids, the proposed central scheme can handle adapted cells/grids. To have cost effective and accurate simulations, the multiresolution–based grid adaptation approach is then integrated with the RBF-based central scheme. Effects of condition numbers of RBFs, computational complexity and cost of the proposed scheme are studied. Finally, different 1-D coupled thermoelasticity benchmarks are presented. There, performance of the adaptive RBF-based formulation is compared with that of the adaptive Kurganov-Tadmor (KT) second-order central high-resolution scheme with the total variation diminishing (TVD) property.

A cell centered high resolution scheme with the dissipation term based on dominant wave approach is examined here. A simplied dominant eigenvalue expression based on the momentum equations together with low Mach number precondtioning is introduced. The dominant wave scheme was implemented in the unstructured cell centered framework capable of supporting polyhedral meshes. Second order of accuracy is achieved by using slope limiters for the primitive data reconstructed at the nite volume cell faces. The main advantage of the scheme is the resulting simplied dissipation term that does not require computation of eigenvectors while retaining the high resolution capabilities. It is shown that the dominant wave scheme has similar resolution to the ux dierence upwind scheme based on the Roe Riemann solver. In addition, the dominant wave scheme has similar resolution characteristics compared to other central schemes such as the Kurganov-Tadmor scheme, and comparison to the HLLC scheme showed similar resolution capabilities. A number of computational tests in 1D, 2D and 3D are performed in order to assess the resolution capabilities of the scheme.

Subject index to volumes 74–80, 1996

We construct, analyze, and implement a new nonoscillatory high-resolution scheme for two-dimensional hyperbolic conservation laws. The scheme is a predictor-corrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewise-linear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a second-order, nonoscillatory central scheme, a natural extension of the one-dimensional second-order central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408--448]. As in the one-dimensional case, the main feature of our two-dimensional scheme is simplicity. In particular, this central scheme does not require the intricate and time-consuming (approximate) Riemann solvers which are essential for the high-resolution upwind schemes; in fact, even the computation of the exact Jacobians can be avoided. Moreover, the central scheme is genuinely multidimensional in the sense that it does not necessitate dimensional splitting. We prove that the scheme satisfies the scalar maximum principle, and in the more general context of systems, our proof indicates that the scheme is positive (in the sense of Lax and Liu [CFD Journal, 5 (1996), pp. 1--24]). We demonstrate the application of our central scheme to several prototype two-dimensional Euler problems. Our numerical experiments include the resolution of shocks oblique to the computational grid; they show how our central scheme solves with high resolution the intricate wave interactions in the so-called double Mach reflection problem [J. Comput. Phys., 54 (1988), pp. 115--173] without following the characteristics; and finally we report on the accurate ray solutions of a weakly hyperbolic system [J. Comput. Appl. Math., 74 (1996), pp. 175--192], rays which otherwise are missed by the dimensional splitting approach. Thus, a considerable amount of simplicity and robustness is gained while achieving stability and high resolution.

High resolution central scheme using a new upwind slope limiter for hyperbolic conservation laws

Developing high resolution finite difference scheme and enabling the use of this scheme on complex geometry are the aims of this study. High resolution has been achieved by Dissipative Compact Schemes (DCS), however, according to the recent research, applications of DCS on complex geometry may have serious problem for that the Geometric Conservation Law (GCL) is not satisfied, and this may cause numerical instability. To cope with this problem, a new scheme named Hybrid cell-edge and cell-node Dissipative Compact Scheme (HDCS) has been formulated. The formulation of the HDCS contains two steps. First, a new central compact scheme is formulated for the purpose of conveniently fulfilling the GCL, and then dissipation is added on the central scheme by high-order dissipative interpolation of cell-edge variables. The solutions of Euler and Navier-Stokes equations show that the HDCS can be applied successfully on complex geometry, while the DCS may suffer numerical instabilities. Moreover, high resolution of the HDCS may be observed in the test of scattering of acoustic waves by multiple cylinders.

A well-balanced high-resolution shape-preserving central scheme to solve one-dimensional sediment transport equations

New High-Resolution Semi-discrete Central Schemes for Hamilton–Jacobi Equations

We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408--463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892--1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.

PurposeThis paper aims to numerically study the compositional flow of two- and three-phase fluids in one-dimensional porous media and to make a comparison between several upwind and central numerical schemes.Design/methodology/approachImplicit pressure explicit composition (IMPEC) procedure is used for discretization of governing equations. The pressure equation is solved implicitly, whereas the mass conservation equations are solved explicitly using different upwind (UPW) and central (CEN) numerical schemes. These include classical upwind (UPW-CLS), flux-based decomposition upwind (UPW-FLX), variable-based decomposition upwind (UPW-VAR), Roe’s upwind (UPW-ROE), local Lax–Friedrichs (CEN-LLF), dominant wave (CEN-DW), Harten–Lax–van Leer (HLL) and newly proposed modified dominant wave (CEN-MDW) schemes. To achieve higher resolution, high-order data generated by either monotone upstream-centered schemes for conservation laws (MUSCL) or weighted essentially non-oscillatory (WENO) reconstructions are used.FindingsIt was found that the new CEN-MDW scheme can accurately solve multiphase compositional flow equations. This scheme uses most of the information in flux function while it has a moderate computational cost as a consequence of using simple algebraic formula for the wave speed approximation. Moreover, numerically calculated wave structure is shown to be used as a tool for a priori estimation of problematic regions, i.e. degenerate, umbilic and elliptic points, which require applying correction procedures to produce physically acceptable (entropy) solutions.Research limitations/implicationsThis paper is concerned with one-dimensional study of compositional two- and three-phase flows in porous media. Temperature is assumed constant and the physical model accounts for miscibility and compressibility of fluids, whereas gravity and capillary effects are neglected.Practical implicationsThe proposed numerical scheme can be efficiently used for solving two- and three-phase compositional flows in porous media with a low computational cost which is especially useful when the number of chemical species increases.Originality/valueA new central scheme is proposed that leads to improved accuracy and computational efficiency. Moreover, to the best of authors knowledge, this is the first time that the wave structure of compositional model is investigated numerically to determine the problematic situations during numerical solution and adopt appropriate correction techniques.

We are concerned with numerical schemes for solving scalar hyperbolic conservation laws arising in the simulation of multiphase flows in heterogeneous porous media. These schemes are non-oscillatory and enjoy the main advantage of Godunov-type central schemes: simplicity, i.e., they employ neither characteristic decomposition nor pproximate Riemann solvers. This makes them universal methods that can be applied to a wide variety of physical problems, including hyperbolic systems. We compare the Kurganov-Tadmor (KT) [1] semi-discrete central scheme with the Nessyahu-Tadmor (NT) [2] central scheme. The KT scheme uses more precise information about the local speeds of propagation together with integration over nonuniform control volumes, which contain the Riemann fans. The numerical dissipation in the (KT) scheme is smaller than in the original NT scheme, however the NT scheme can use larger time steps. Numerical simulations are presented for two-phase flow problems in very heterogeneous formations. We find the KT scheme to be considerably less diffusive, particularly in the presence of viscous fingers, which lead to strong restrictions on the time step selection. REFERENCES [1] Kurganov, A. & Tadmor, E., 2000. New high-resolution central schemes for nonlinear conser- vation laws and convection-diffusion equations. Journal of Computational Physics, vol. 160, pp. 241­282. [2] Nessayahu, H. & Tadmor, E., 1990. Non-oscillatory central differencing for hyperbolic conser- vation laws. Journal of Computational Physics, vol. 87, n. 2, pp. 408­463.