The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids II: Extension to high order finite volume schemes

In this paper, the multidimensional limiter for the second order finite volume schemes on the unstructured grid, namely the Weighted Biased Average procedure developed in our previous paper is extended to high order finite volume schemes solving hyperbolic conservation laws. This extension relies on two key techniques: the secondary reconstruction and the successive limiting procedure. These techniques are discussed in detail in the present paper. Numerical experiments shows that this limiting procedure is very effective in removing numerical oscillations in the vicinity of discontinuities. And furthermore this procedure is efficient, robust and accuracy preserving.     

High order finite volume approximations of differential operators on nonuniform grids

When computing numerical solutions to partial differential equations, difference operators that mimic the crucial properties of the differential operators are usually more accurate than those that do not. Properties such as symmetry, conservation, stability, and the duality relationships and identities between the gradient, curl, and divergence operators are all important. Using the finite volume method, we have derived local, accurate, reliable and efficient difference methods t divergence, gradient, and curl operators are defined using a discrete versions of the divergence theorem and Stokes' theorem. These methods are especially powerful on coarse nonuniform grids and in calculations where the mesh moves to track interfaces or shocks. Numerical examples comparing local second and fourth-order finite volume approximations to conservation laws on very rough grids are used to demonstrate the advantages of the higher order methods.     

A dynamic finite volume scheme for large-eddy simulation on unstructured grids

In recent years there has been considerable progress in the application of large-eddy simulation (LES) to increasingly complex flow configurations. Nevertheless a lot of fundamental problems still need to be solved in order to apply LES in a reliable way to real engineering problems, where typically finite-volume codes on unstructured meshes are used. A self-adaptive discretisation scheme, in the context of an unstructured finite-volume flow solver, is investigated in the case of isotropic turbulence at infinite Reynolds number. The Smagorinsky and dynamic Smagorinsky sub-grid scale models are considered. A discrete interpolation filter is used for the dynamic model. It is one of the first applications of a filter based on the approach presented by Marsden et al. In this work, an original procedure to impose the filter shape through a specific selection process of the basic filters is also proposed. Satisfactory results are obtained using the self-adaptive scheme for implicit LES. When the scheme is coupled with the sub-grid scale models, the numerical dissipation is shown to be dominant over the sub-grid scale component. Nevertheless the effect of the sub-grid scale models appears to be important and beneficial, improving in particular the energy spectra. A test on fully developed channel flow at Re_τ = 395 is also performed, comparing the non-limited scheme with the self-adaptive scheme for implicit LES. Once again the introduction of the limiter proves to be beneficial.     

L-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios

This work is devoted to the design of multi-dimensional finite volume schemes for solving transport equations on unstructured grids. In the framework of MUSCL vertex-based methods we construct numerical fluxes such that the local maximum property is guaranteed under an explicit Courant–Friedrichs–Levy condition. The method can be naturally completed by adaptive local mesh refinements and it turns out that the mesh generation is less constrained than when using the competitive cell-centered methods. We illustrate the effectiveness of the scheme by simulating variable density incompressible viscous flows. Numerical simulations underline the theoretical predictions and succeed in the computation of high density ratio phenomena such as a water bubble falling in air.     

Compact third-order limiter functions for finite volume methods

We consider finite volume methods for the numerical solution of conservation laws. In order to achieve high-order accurate numerical approximation to non-linear smooth functions, we introduce a new class of limiter functions for the spatial reconstruction of hyperbolic equations. We therefore employ and generalize the idea of double-logarithmic reconstruction of Artebrant and Schroll [R. Artebrant, H.J. Schroll, Limiter-free third order logarithmic reconstruction, SIAM J. Sci. Comput. 28 (2006) 359-381].     

Splitting based finite volume schemes for ideal MHD equations

We design finite volume schemes for the equations of ideal magnetohydrodynamics (MHD) and based on splitting these equations into a fluid part and a magnetic induction part. The fluid part leads to an extended Euler system with magnetic forces as source terms. This set of equations are approximated by suitable two- and three-wave HLL solvers. The magnetic part is modeled by the magnetic induction equations which are approximated using stable upwind schemes devised in a recent paper [F. Fuchs, K.H. Karlsen, S. Mishra, N.H. Risebro, Stable upwind schemes for the Magnetic Induction equation. Math. Model. Num. Anal., Available on conservation laws preprint server, submitted for publication, URL: <http://www.math.ntnu.no/conservation/2007/029.html>]. These two sets of schemes can be combined either component by component, or by using an operator splitting procedure to obtain a finite volume scheme for the MHD equations. The resulting schemes are simple to design and implement. These schemes are compared with existing HLL type and Roe type schemes for MHD equations in a series of numerical experiments. These tests reveal that the proposed schemes are robust and have a greater numerical resolution than HLL type solvers, particularly in several space dimensions. In fact, the numerical resolution is comparable to that of the Roe scheme on most test problems with the computational cost being at the level of a HLL type solver. Furthermore, the schemes are remarkably stable even at very fine mesh resolutions and handle the divergence constraint efficiently with low divergence errors.     

Geometric conservation law and applications to high-order finite difference schemes with stationary grids

The geometric conservation law (GCL) includes the volume conservation law (VCL) and the surface conservation law (SCL). Though the VCL is widely discussed for time-depending grids, in the cases of stationary grids the SCL also works as a very important role for high-order accurate numerical simulations. The SCL is usually not satisfied on discretized grid meshes because of discretization errors, and the violation of the SCL can lead to numerical instabilities especially when high-order schemes are applied. In order to fulfill the SCL in high-order finite difference schemes, a conservative metric method (CMM) is presented. This method is achieved by computing grid metric derivatives through a conservative form with the same scheme applied for fluxes. The CMM is proven to be a sufficient condition for the SCL, and can ensure the SCL for interior schemes as well as boundary and near boundary schemes. Though the first-level difference operators δ₃ have no effects on the SCL, no extra errors can be introduced as δ₃ = δ₂. The generally used high-order finite difference schemes are categorized as central schemes (CS) and upwind schemes (UPW) based on the difference operator δ₁ which are used to solve the governing equations. The CMM can be applied to CS and is difficult to be satisfied by UPW. Thus, it is critical to select the difference operator δ₁ to reduce the SCL-related errors. Numerical tests based on WCNS-E-5 show that the SCL plays a very important role in ensuring free-stream conservation, suppressing numerical oscillations, and enhancing the robustness of the high-order scheme in complex grids.     

Compact upwind schemes on adaptive octrees

Compact high-order upwind schemes using reconstruction from cell-averages are derived for application with the compressible three-dimensional Navier–Stokes equations. An adaptive-octree mesh, combined with the Adams–Bashforth–Moulton family of predictor–corrector schemes, provides a conservative high-order time-integration platform supporting localized h-refinement and timestep sub-cycling. Numerical examples for smooth flows demonstrate the improvement over explicit upwind schemes and formal accuracy of the schemes, as well as the behavior in wall-bounded regions, and the resolution of a broad wavenumber spectrum.     

A coupled finite volume solver for the solution of incompressible flows on unstructured grids

This paper reports on a newly developed fully coupled pressure-based algorithm for the solution of laminar incompressible flow problems on collocated unstructured grids. The implicit pressure-velocity coupling is accomplished by deriving a pressure equation in a procedure similar to a segregated SIMPLE algorithm using the Rhie–Chow interpolation technique and assembling the coefficients of the momentum and continuity equations into one diagonally dominant matrix. The extended systems of continuity and momentum equations are solved simultaneously and their convergence is accelerated by using an algebraic multigrid solver. The performance of the coupled approach as compared to the segregated approach, exemplified by SIMPLE, is tested by solving five laminar flow problems using both methodologies and comparing their computational costs. Results indicate that the number of iterations needed by the coupled solver for the solution to converge to a desired level on both structured and unstructured meshes is grid independent. For relatively coarse meshes, the CPU time required by the coupled solver on structured grid is lower than the CPU time required on unstructured grid. On dense meshes however, this is no longer true. For low and moderate values of the grid aspect ratio, the number of iterations required by the coupled solver remains unchanged, while the computational cost slightly increases. For structured and unstructured grid systems, the required number of iterations is almost independent of the grid size at any value of the grid expansion ratio. Recorded CPU time values show that the coupled approach substantially reduces the computational cost as compared to the segregated approach with the reduction rate increasing as the grid size increases.     

Increasing the accuracy in locally divergence-preserving finite volume schemes for MHD

It is of utmost interest to control the divergence of the magnetic flux in simulations of the ideal magnetohydrodynamic equations since, in general, divergence errors tend to accumulate and render the schemes unstable. This paper presents a higher-order extension of the locally divergence-preserving procedure developed in Torrilhon [M. Torrilhon, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations, SIAM J. Sci. Comput. 26 (2005) 1166–1191]; a fourth-order accurate local redistribution of the numerical magnetic field fluxes of a finite volume base scheme is introduced. The redistribution ensures that a fourth-order accurate discrete divergence operator is preserved to round off errors when applied to the cell averages of the magnetic flux density. The developed procedure is applicable to generic semi-discrete finite volume schemes and its purpose is to stabilize the schemes using a local procedure that respects the accuracy of the base scheme to a greater extent than the previous second-order achievements. Numerical experiments that demonstrate the properties of the new procedure are also presented.     

Wavenumber-extended high-order oscillation control finite volume schemes for multi-dimensional aeroacoustic computations

A new numerical method toward accurate and efficient aeroacoustic computations of multi-dimensional compressible flows has been developed. The core idea of the developed scheme is to unite the advantages of the wavenumber-extended optimized scheme and M-AUSMPW+/MLP schemes by predicting a physical distribution of flow variables more accurately in multi-space dimensions. The wavenumber-extended optimization procedure for the finite volume approach based on the conservative requirement is newly proposed for accuracy enhancement, which is required to capture the acoustic portion of the solution in the smooth region. Furthermore, the new distinguishing mechanism which is based on the Gibbs phenomenon in discontinuity, between continuous and discontinuous regions is introduced to eliminate the excessive numerical dissipation in the continuous region by the restricted application of MLP according to the decision of the distinguishing function. To investigate the effectiveness of the developed method, a sequence of benchmark simulations such as spherical wave propagation, nonlinear wave propagation, shock tube problem and vortex preservation test problem are executed. Also, throughout more realistic shock–vortex interaction and muzzle blast flow problems, the utility of the new method for aeroacoustic applications is verified by comparing with the previous numerical or experimental results.     

A nominally second-order accurate finite volume cell-centered scheme for anisotropic diffusion on two-dimensional unstructured grids

In this paper, we describe a second-order accurate cell-centered finite volume method for solving anisotropic diffusion on two-dimensional unstructured grids. The resulting numerical scheme, named CCLAD (Cell-Centered LAgrangian Diffusion), is characterized by a local stencil and cell-centered unknowns. It is devoted to the resolution of diffusion equation on distorted grids in the context of Lagrangian hydrodynamics wherein a strong coupling occurs between gas dynamics and diffusion. The space discretization relies on the introduction of two half-edge normal fluxes and two half-edge temperatures per cell interface using the partition of each cell into sub-cells. For each cell, the two half-edge normal fluxes attached to a node are expressed in terms of the half-edge temperatures impinging at this node and the cell-centered temperature. This local flux approximation can be derived through the use of either a sub-cell variational formulation or a finite difference approximation, leading to the two variants CCLADS and CCLADNS. The elimination of the half-edge temperatures is performed locally at each node by solving a small linear system which is obtained by enforcing the continuity condition of the normal heat flux across sub-cell interface impinging at the node. The accuracy and the robustness of the present scheme is assessed by means of various numerical test cases.     

A collocated finite volume scheme to solve free convection for general non-conforming grids

We present a new collocated numerical scheme for the approximation of the Navier–Stokes and energy equations under the Boussinesq assumption for general grids, using the velocity–pressure unknowns. This scheme is based on a recent scheme for the diffusion terms. Stability properties are drawn from particular choices for the pressure gradient and the non-linear terms. Convergence of the approximate solutions may be proven mathematically. Numerical results show the accuracy of the scheme on irregular grids.     

Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes

We consider a non-linear finite volume (FV) scheme for stationary diffusion equation. We prove that the scheme is monotone, i.e. it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients. The scheme is extended to regular star-shaped polygonal meshes and isotropic heterogeneous coefficients.     

Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors

We consider a family of explicit one-step time discretizations for finite volume and discontinuous Galerkin schemes, which is based on a predictor-corrector formulation. The predictor remains local taking into account the time evolution of the data only within the grid cell. Based on a space–time Taylor expansion, this idea is already inherent in the MUSCL finite volume scheme to get second order accuracy in time and was generalized in the context of higher order ENO finite volume schemes. We interpret the space–time Taylor expansion used in this approach as a local predictor and conclude that other space–time approximate solutions of the local Cauchy problem in the grid cell may be applied. Three possibilities are considered in this paper: (1) the classical space–time Taylor expansion, in which time derivatives are obtained from known space-derivatives by the Cauchy–Kovalewsky procedure; (2) a local continuous extension Runge–Kutta scheme and (3) a local space–time Galerkin predictor with a version suitable for stiff source terms. The advantage of the predictor–corrector formulation is that the time evolution is done in one step which establishes optimal locality during the whole time step. This time discretization scheme can be used within all schemes which are based on a piecewise continuous approximation as finite volume schemes, discontinuous Galerkin schemes or the recently proposed reconstructed discontinuous Galerkin or P_NP_M schemes. The implementation of these approaches is described, advantages and disadvantages of different predictors are discussed and numerical results are shown.     

On the relation between finite element and finite volume schemes for compressible flows with cylindrical and spherical symmetry

A complete set of equivalence conditions, connecting the node-centered finite volume and the mass-lumped finite element schemes, is derived for the first time for the one-dimensional compressible Euler equations with cylindrical and spherical symmetry. Analytical expressions for the evaluation of the equivalent cell volumes and interface normals in terms of the finite element integrals are presented. Numerical experiments for compressible unsteady flows, including expanding and converging shock problems, are carried out using the new approach and the differences with the results from a finite volume scheme violating the equivalence conditions are discussed.     

High order well-balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres

Wave propagation in idealized stellar atmospheres is modeled by the equations of ideal MHD, together with the gravity source term. The waves are modeled as small perturbations of isothermal steady states of the system. We consider a formulation of ideal MHD based on the Godunov–Powell form, with an embedded potential magnetic field appearing as a parameter. The equations are discretized by finite volume schemes based on approximate Riemann solvers of the HLL type and upwind discretizations of the Godunov–Powell source terms. Local hydrostatic reconstructions and suitable discretization of the gravity source term lead to a well-balanced scheme, i.e., a scheme which exactly preserves a discrete version of the relevant steady states. Higher order of accuracy is obtained by employing suitable minmod, ENO and WENO reconstructions, based on the equilibrium variables, to construct a well-balanced scheme. The resulting high order well-balanced schemes are validated on a suite of numerical experiments involving complex magnetic fields. The schemes are observed to be robust and resolve the complex physics well.     

High order finite volume methods on wavelet-adapted grids with local time-stepping on multicore architectures for the simulation of shock-bubble interactions

We present a space–time adaptive solver for single- and multi-phase compressible flows that couples average interpolating wavelets with high-order finite volume schemes. The solver introduces the concept of wavelet blocks, handles large jumps in resolution and employs local time-stepping for efficient time integration. We demonstrate that the inherently sequential wavelet-based adaptivity can be implemented efficiently in multicore computer architectures using task-based parallelism and introducing the concept of wavelet blocks. We validate our computational method on a number of benchmark problems and we present simulations of shock-bubble interaction at different Mach numbers, demonstrating the accuracy and computational performance of the method.     

Hierarchical reconstruction for spectral volume method on unstructured grids

The hierarchical reconstruction (HR) [Y.-J. Liu, C.-W. Shu, E. Tadmor, M.-P. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal. 45 (2007) 2442–2467; Z.-L. Xu, Y.-J. Liu, C.-W. Shu, Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO type linear reconstruction and partial neighboring cells, J. Comput. Phys. 228 (2009) 2194–2212] is applied to a piecewise quadratic spectral volume method on two-dimensional unstructured grids as a limiting procedure to prevent spurious oscillations in numerical solutions. The key features of this HR are that the reconstruction on each control volume only uses adjacent control volumes, which forms a compact stencil set, and there is no truncation of higher degree terms of the polynomial. We explore a WENO-type linear reconstruction on each hierarchical level for the reconstruction of high degree polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed. We demonstrate that the hierarchical reconstruction can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions.