High-order Finite Volume Research Articles

High-order upwind finite volume element schemes for modelling of neuronal firing

In this paper, we design high-order upwind finite volume element method (UFVEM) schemes to solve first-order hyperbolic partial differential-difference equation with shift, which arises in the modelling of neuronal firing. Error analysis for the schemes shows that the approximate solution obtained from UFVEM schemes is in the L2 norm. Numerical examples are also provided to support the method and the theoretical analysis. The new schemes preserve the height of neuronal impulses better than usual Lax–Friedrichs schemes which we have shown in numerical examples. In addition, the numerical oscillation produced by implicit centre finite difference schemes can also be eliminated.

Parametrized Maximum Principle Preserving Flux Limiters for High Order Schemes Solving Multi-Dimensional Scalar Hyperbolic Conservation Laws

In this paper, we will extend the strict maximum principle preserving flux limiting technique developed for one dimensional scalar hyperbolic conservation laws to the two-dimensional scalar problems. The parametrized flux limiters and their determination from decoupling maximum principle preserving constraint is presented in a compact way for two-dimensional problems. With the compact fashion that the decoupling is carried out, the technique can be easily applied to high order finite difference and finite volume schemes for multi-dimensional scalar hyperbolic problems. For the two-dimensional problem, the successively defined flux limiters are developed for the multi-stage total-variation-diminishing Runge---Kutta time-discretization to improve the efficiency of computation. The high order schemes with successive flux limiters provide high order approximation and maintain strict maximum principle with mild Courant-Friedrichs-Lewy constraint. Two dimensional numerical evidence is given to demonstrate the capability of the proposed approach.

Higher-order finite volume methods II: Inf–sup condition and uniform local ellipticity

The main purpose of this paper is to study the construction of higher-order finite volume methods (FVMs) of triangle meshes. We investigate the relationship of the three theoretical notions crucial in the construction of FVMs: the uniform ellipticity of the family of its discrete bilinear forms, its inf–sup condition and its uniform local ellipticity. Both the uniform ellipticity of the family of the discrete bilinear forms and its inf–sup condition guarantee the unique solvability of the FVM equations and the optimal error estimate of the approximate solution. We characterize the uniform ellipticity in terms of the inf–sup condition and a geometric condition on the bijective operator mapping from the trial space onto the test space involved in the construction of FVMs. We present a geometric interpretation of the inf–sup condition. Moreover, since the uniform local ellipticity is a convenient sufficient condition for the uniform ellipticity, we further provide sufficient conditions and necessary conditions of the uniform local ellipticity.

A diffuse interface method for complex three-dimensional free surface flows

In this article the weakly compressible two-phase diffuse interface method (DIM) for the simulation of complex two-dimensional non-hydrostatic free surface flows proposed by Dumbser in [24] is extended to three-dimensional unstructured tetrahedral meshes. As in the 2D case, a reduced version of the Baer–Nunziato model for compressible multiphase flows is used. The physical model is closed by the Tait equation of state for water and can be implemented easily into existing compressible codes based on high resolution shock capturing finite volume schemes. Since the proposed model is fully three-dimensional, it includes the fluid accelerations in gravity direction and hence does not assume a hydrostatic pressure distribution, like the classical shallow water equations. Furthermore, the 3D two-phase model can naturally deal also with breaking waves. To solve the system of conservation laws of mass and momentum coupled with the non-conservative evolution equation of the fluid volume fraction, a high order path-conservative one-step WENO finite volume scheme is applied, together with a new generalized Osher-type Riemann solver at the element interfaces. The accurate Riemann solver in combination with a high order finite volume approach leads to a simple but sharp resolution of the free surface.A thorough comparison of experimental reference data with the computational results obtained for a large set of three-dimensional test cases shows the suitability of the present approach for the accurate simulation of complex three-dimensional free surface flows. The use of a compressible flow model allows the method to simulate both, low speed and high speed free surface flow problems, which makes the approach applicable to a very wide class of environmental and industrial free surface flow problems.

Deterministic unsteady and aeroelastic flow simulations with high-order FVM schemes

During the last few years, unsteady and aeroelastic simulations have become a standard part of CFD computations in aeronautical applications. They are usually used with various types of turbulent models the most of which are derived from system of the Reynolds-Averaged Navier–Stokes equations (RANS). However, the RANS system itself is derived using time-averaging and hence it neglects certain type of unsteadiness in its own nature. Therefore it is necessary to investigate what is the ability of the contemporary numerical methods (i.e. combinations of numerical schemes and turbulence models) to simulate various types of unsteady flows. The authors carried out a series of numerical tests of two types of unsteady flows in external aerodynamics. Namely the transonic compressible flow over harmonically oscillating NACA 0012 aerofoil and subsonic compressible flow over the same aerofoil with oscillations induced by the flow itself (i.e. considering the aeroelastic interaction). Three different high-order finite volume schemes with various contemporary turbulence models were chosen for the testing. In the first researched case the computations were carried out using: the Modified Causon’s scheme “as reported by Furmanek and Kozel (Kybernetika 45(4):567–579, 2009)” (derived from TVD form of the classical MacCormack scheme) and the solver edge developed by the Swedish defence and research agency FOI (n-stage Runge–Kutta scheme, central differences with Jameson’s artificial dissipation, geometric multigrid, implicit residual smoothing). The tested models of turbulence were: the one-equation Spalart–Allmaras model, the two-equation Kok’s TNT model and the EARSM model. Unfortunately, none of these models produced results able to serve as a reliable numerical simulation of the first researched case. The main cause is probably the interaction between the turbulent boundary layer and shock-wave formed on the top side of the aerofoil. For the purpose of further investigation the Modified Causon’s scheme was adapted for modelling aeroelastic interaction between aerofoil and flow with one degree of freedom. Results of this simulation correspond better to the real flow, but it is necessary to carry out another tests to find out what is the exact cause of the observed problems.

OpenCL Implementation of Basic Operations for a High-order Finite-volume Polynomial Scheme on Unstructured Hybrid Meshes

A parallel finite-volume algorithm based on a cell-centered high-order polynomial scheme for unstructured hybrid meshes is under consideration. The work is focused on the adaptation and optimization of basic operations of the algorithm to different architec- tures of massively-parallel accelerators including GPU of AMD and NVIDIA. Such an algorithm is especially problematic for the GPU architectures since it has very low FLOP per byte ratio meaning that performance is dominated by the memory bandwidth but not the computing performance of a device. At the same time it has irregular memory access pattern since unstructured meshes are used. The calculation of polynomial coefficients and the calculation of convective fluxes through faces of cells are the most interesting and time consuming operations of the algorithm. Implementations of these operations for accelerators using OpenCL are considered here in detail. The ways to improve the computational efficiency are proposed, performance measurement results reaching up to 160 GFLOPS on a single GPU device are demonstrated.

A high order finite volume element method for solving elliptic partial integro-differential equations

In this paper, we develop a finite volume element method of order p for solving elliptic integro-differential equations in two dimensions. These types of equations arise in questions of hereditary phenomena in physics. The H1 norm error estimates are discussed, the convergence result in H1 norm is proved and some numerical results are studied to illustrate the effectiveness of the method.

GPU-based flow simulation with detailed chemical kinetics

The current paper reports on the implementation of a numerical solver on the Graphic Processing Units (GPUs) to model reactive gas mixtures with detailed chemical kinetics. The solver incorporates high-order finite volume methods for solving the fluid dynamical equations coupled with stiff source terms. The chemical kinetics are solved implicitly via an operator-splitting method. We explored different approaches in implementing a fast kinetics solver on the GPU. The detail of the implementation is discussed in the paper. The solver is tested with two high-order shock capturing schemes: MP5 (Suresh and Huynh, 1997) [9] and ADERWENO (Titarev and Toro, 2005) [10]. Considering only the fluid dynamics calculation, the speed-up factors obtained are 30 for the MP5 scheme and 55 for ADERWENO scheme. For the fully-coupled solver, the performance gain depended on the size of the reaction mechanism. Two different examples of chemistry were explored. The first mechanism consisted of 9 species and 38 reactions, resulting in a speed-up factor up to 35. The second, larger mechanism, consisted of 36 species and 308 reactions, resulting in a speed-up factor of up to 40.

A new locally conservative numerical method for two-phase flow in heterogeneous poroelastic media

We construct a new class of locally conservative numerical methods for two-phase immiscible flow in heterogeneous poroelastic media. Within the framework of the so-called iteratively coupled methods and fixed-stress split algorithm we develop mixed finite element methods for the flow and geomechanics subsystems which furnish locally conservative Darcy velocity and transient porosity input fields for the transport problem for the water saturation. Such hyperbolic equation is decomposed within an operator splitting technique based on a predictor–corrector scheme with the predictor step discretized by a higher-order non-oscillatory finite volume central scheme. The proposed scheme adopts an inhomogeneous dual mesh with variable cell size ruled by the local wave speed of propagation to compute numerical fluxes at cell edges. In the limit of small time steps the central scheme gives rise to a semidiscrete formulation for the water saturation capable of incorporating heterogeneous porosity fields and generalized flux functions including the water transport due to the solid phase velocity. Numerical simulations of a water-flooding problem in secondary oil recovery are presented for different realizations of the input random fields (permeability, Young modulus and initial porosity). Comparison between the accuracies of the proposed approach and the traditional one-way coupled hydro-geomechanical formulation are presented. The effects of the cross-correlation between the input random fields and compaction drive mechanism upon finger growth and breakthrough curves are also analyzed. A notable feature of the formulation proposed herein is the accurate prediction of the influence of geomechanical effects upon the unstable movement of the water front, whose evolution is dictated by rock heterogeneity and unfavorable viscosity ratio, without deteriorating the local conservative character of the numerical schemes.

A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems

This paper deals with the formulation of a semi-intrusive (SI) method allowing the computation of statistics of linear and non linear PDEs solutions. This method shows to be very efficient to deal with probability density function of whatsoever form, long-term integration and discontinuities in stochastic space.Given a stochastic PDE where randomness is defined on Ω, starting from (i) a description of the solution in term of a space variables, (ii) a numerical scheme defined for any event ω∈Ω and (iii) a (family) of random variables that may be correlated, the solution is numerically described by its conditional expectancies of point values or cell averages and its evaluation constructed from the deterministic scheme. One of the tools is a tessellation of the random space as in finite volume methods for the space variables. Then, using these conditional expectancies and the geometrical description of the tessellation, a piecewise polynomial approximation in the random variables is computed using a reconstruction method that is standard for high order finite volume space, except that the measure is no longer the standard Lebesgue measure but the probability measure. This reconstruction is then used to formulate a scheme on the numerical approximation of the solution from the deterministic scheme. This new approach is said semi-intrusive because it requires only a limited amount of modification in a deterministic solver to quantify uncertainty on the state when the solver includes uncertain variables.The effectiveness of this method is illustrated for a modified version of Kraichnan–Orszag three-mode problem where a discontinuous pdf is associated to the stochastic variable, and for a nozzle flow with shocks. The results have been analyzed in terms of accuracy and probability measure flexibility. Finally, the importance of the probabilistic reconstruction in the stochastic space is shown up on an example where the exact solution is computable, the viscous Burgers equation.

Moving Kriging reconstruction for high-order finite volume computation of compressible flows

This paper describes the development of a high-order finite volume method for the solution of compressible viscous flows on unstructured meshes. The novelty of this approach is based on the use of moving Kriging shape functions for the computation of the derivatives in the numerical flux reconstruction step at the cell faces. For each cell, the successive derivatives of the flow variables are deduced from the interpolation function constructed from a compact stencil support for both Gaussian and quartic spline correlation models. A particular attention is paid for the study of the influence of the correlation parameter onto the accuracy of the numerical scheme. The effect of the size of the moving Kriging stencil is also investigated. Robustness and convergence properties are studied for various inviscid and viscous flows. Results reveal that the moving Kriging shape function can be considered as an interesting alternative for the development of high-order methodology for complex geometries.

An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations

We present a high-order well-balanced unstructured finite volume (FV) scheme on triangular meshes for modeling weakly nonlinear and weakly dispersive water waves over slowly varying bathymetries, as described by the 2D depth-integrated extended Boussinesq equations of Nwogu, rewritten here in conservation law form. The FV scheme numerically solves the conservative form of the equations following the median dual node-centered approach, for both the advective and dispersive part of the equations. For the advective fluxes, the scheme utilizes an approximate Riemann solver along with a well-balanced topography source term upwinding. Higher order accuracy in space and time is achieved through a MUSCL-type reconstruction technique and through a strong stability preserving explicit Runge–Kutta time stepping. Special attention is given to the accurate numerical treatment of moving wet/dry fronts and boundary conditions. The model is applied to several examples of non-breaking wave propagation over variable topographies and the computed solutions are compared to experimental data. The presented results indicate that the presented FV model is robust and capable of simulating wave transformations from relatively deep to shallow water, providing accurate predictions of the wave's propagation, shoaling and runup.

Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials

This paper extends the MOOD method proposed by the authors in [A high-order finite volume method for hyperbolic systems: Multi-Dimensional Optimal Order Detection (MOOD). J Comput Phys 2011;230:4028–50], along two complementary axes: extension to very high-order polynomial reconstruction on non-conformal unstructured meshes and new detection criteria. The former is a natural extension of the previous cited work which confirms the good behavior of the MOOD method. The latter is a necessary brick to overcome limitations of the discrete maximum principle used in the previous work. Numerical results on advection problems and hydrodynamics Euler equations are presented to show that the MOOD method is effectively high-order (up to sixth-order), intrinsically positivity-preserving on hydrodynamics test cases and computationally efficient.

An Analysis of the Dissipation and Dispersion Errors of the P N P M Schemes

We examine the dispersion and dissipation properties of the P N P M schemes for linear wave propagation problems. P N P M scheme are based on P N discontinuous Galerkin base approximations augmented with a cell centered polynomial least-squares reconstruction from degree N up to the design polynomial degree M. This methodology can be seen as a generalized discretization framework, as cell centered high order finite volume schemes (N=0) and discontinuous Galerkin schemes (N=M) are included as special cases. We show that with respect to the dispersion error, the pure discontinuous Galerkin variant gives typically the best accuracy for a defined number of points per wavelength. Regarding the dissipation behavior, combinations of N and M exist that result in slightly lower errors for a given resolution. An investigation of the influence of the stencil size on the accuracy of the scheme shows that the errors are smaller the smaller the stencil size for the reconstruction.

ADER scheme on unstructured meshes for shallow water: simulation of tsunami waves

SUMMARY In this paper, we extend Arbitrary accuracy DErivatives Riemann (ADER)-type finite volume numerical methods to simulate tsunami wave propagation and steady state flows with geometrical sources. The mathematical formulation of the physical problem is based on the shallow water equations with non-constant bathymetry. Although, the physical process described by the shallow water equations preserves the steady state condition balancing fluxes and sources, a number of finite volumes numerical methods fail to preserve this ‘well-balanced’ property at the discrete level. In the last few years many numerical methods have been presented that fulfil this condition although retaining high-order accuracy. However, most of them are based on a 1-D quadrature rule and are suitable only for Cartesian grids. Here we present an arbitrary high-order finite volume scheme based on the ADER approach on 2-D unstructured meshes that preserves steady state solutions with arbitrarily high order of accuracy. This property allows us to properly represent tsunami wave propagation over realistic bathymetries.

A Review of Residual Distribution Schemes for Hyperbolic and Parabolic Problems: The July 2010 State of the Art

AbstractWe describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil. We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes, show that their are really non oscillatory. We also discuss the extension to these methods to parabolic problems. We also draw some research perspectives.

Development of a high-order finite volume method with multiblock partition techniques

This work deals with a new numerical methodology to solve the Navier-Stokes equations based on a finite volume method applied to structured meshes with co-located grids. High-order schemes used to approximate advective, diffusive and non-linear terms, connected with multiblock partition techniques, are the main contributions of this paper. Combination of these two techniques resulted in a computer code that involves high accuracy due the high-order schemes and great flexibility to generate locally refined meshes based on the multiblock approach. This computer code has been able to obtain results with higher or equal accuracy in comparison with results obtained using classical procedures, with considerably less computational effort.

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.

Maximum-principle-satisfying High Order Finite Volume Weighted Essentially Nonoscillatory Schemes for Convection-diffusion Equations

To easily generalize the maximum-principle-satisfying schemes for scalar conservation laws in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 3091-3120] to convection diffusion equations, we propose a nonconventional high order finite volume weighted essentially nonoscillatory (WENO) scheme which can be proved maximum-principle-satisfying. Two-dimensional extensions are straightforward. We also show that the same idea can be used to construct high order schemes preserving the maximum principle for two-dimensional incompressible Navier-Stokes equations in the vorticity stream-function formulation. Numerical tests for the fifth order WENO schemes are reported.

Central Schemes for Nonconservative Hyperbolic Systems

In this work we present a new approach to the construction of high order finite volume central schemes on staggered grids for general hyperbolic systems, including those not admitting a conservation form. The method is based on finite volume space discretization on staggered cells, central Runge--Kutta time discretization, and integration over a family of paths, associated to the system itself, for the generalization of the method to nonconservative systems. Applications to the one- and two-layer shallow water models as prototypes of systems of balance laws and systems with source terms and nonconservative products, respectively, will be illustrated.