High-order Finite Volume Method Research Articles

High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids

We present an approach to solving hyperbolic conservation laws by finite-volume methods on mapped multiblock grids, extending the approach of Colella, Dorr, Hittinger, and Martin (2011) [10] for grids with a single mapping. We consider mapped multiblock domains for mappings that are conforming at inter-block boundaries. By using a smooth continuation of the mapping into ghost cells surrounding a block, we reduce the inter-block communication problem to finding an accurate, robust interpolation into these ghost cells from neighboring blocks. We demonstrate fourth-order accuracy for the advection equation for multiblock coordinate systems in two and three dimensions.

A Three-Dimensional Conservative Coupling Method Between an Inviscid Compressible Flow and a Moving Rigid Solid

We present a conservative method for the three-dimensional coupling between an inviscid compressible flow and a moving rigid solid. We consider an inviscid Euler fluid in conservative form discretized using a high-order monotonicity-preserving finite volume method with a directional operator splitting. An immersed boundary technique is employed through the modification of the finite volume fluxes in the vicinity of the solid. The method yields exact conservation of mass, momentum, and energy of the system, and also exhibits important consistency properties, such as conservation of uniform movement of both fluid and solid as well as the absence of numerical roughness on a straight boundary. The coupling scheme evaluates the fluxes on the fluid side and the forces and torques on the solid side only once every time step, ensuring the computational efficiency of the coupling. We present numerical results assessing the robustness of the method in the case of rigid solids with large displacements.

High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes

In this paper, we generalize the maximum-principle-preserving (MPP) flux limiting technique developed by Xu (2013) [20] to a class of high order finite volume weighted essentially non-oscillatory (WENO) schemes for scalar conservation laws and the compressible Euler system on unstructured meshes in one and two dimensions. The key idea of this parameterized limiting technique is to limit the high order numerical flux with a first order flux which preserves the MPP or positivity-preserving (PP) property. The main purpose of this paper is to investigate the flux limiting approach with high order finite volume method on unstructured meshes which are often needed for solving some important problems on irregular domains. Truncation error analysis based on one-dimensional nonuniform meshes is presented to justify that the proposed MPP schemes can maintain third order accuracy in space and time. We also demonstrate through smooth test problems that the proposed third order MPP/PP WENO schemes coupled with a third order Runge–Kutta (RK) method attain the desired order of accuracy. Several test problems containing strong shocks and complex domain geometries are also presented to assess the performance of the schemes.

Extrapolated local radial basis function collocation method for shallow water problems

This paper provides a novel meshfree numerical method – the extrapolated local-radial-basis-function collocation method (ELRBFCM) – to solve the well-known shallow-water equations (or de Saint-Venant equations). To promote the accuracy and stability of the local-radial-basis-function collocation method (LRBFCM), the extrapolation method is employed in the ELRBFCM. The numerical experiments include the simulation of two linear and three non-linear problems. For the linear problems, the results of the extrapolation method not only have good agreement with analytical solutions, but also show better efficiency than the Euler and RK4 scheme. It is convinced that by the extrapolation process, the approximating order and stability of time integration is automatically controlled. For the non-linear problems, the solutions by the proposed method capture the physical reaction that also obtained in the benchmark by a complex high-order finite-volume method. As a result, it is concluded that even for very long period, the meshfree ELRBFCM with high-order temporal approximation can provide robust, high accurate and great efficient numerical simulation for the shallow-water problems.

A high-order Cartesian-grid finite-volume method for aeroacoustics simulations

A moving-least-square based finite-volume method is developed to simulate acoustic wave propagation and scattering from complicated solid geometries. This hybrid method solves the linearized perturbed compressible equations as the governing equations of the acoustic field. The solid boundaries are embedded in a uniform Cartesian grid and represented using level set fields. Thus, the current approach avoids unstructured grid generation for the irregular geometries. The desired boundary conditions are imposed sharply on the immersed boundaries using a ghost fluid method. The scope of the implementation of the moving moving-least-square approach in the current solver is threefold: reconstruction of the field variables on cell faces for high-order flux construction, population of the ghost cells based on the desired boundary condition, and filtering the high wave number modes near the immersed boundaries. The computational stencils away from the boundaries are identical; hence, only one moving-least-square shape-function is computed and stored with its underlying grid pattern for all the interior cells. This feature significantly reduces the memory requirement of the acoustic solver compared to similar finite-volume method on irregular unstructured mesh. The acoustic solver is validated against several benchmark problems.

A very high-order finite volume method for the time-dependent convection–diffusion problem with Butcher Tableau extension

The time discretization of a very high-order finite volume method may give rise to new numerical difficulties resulting into accuracy degradations. Indeed, for the simple one-dimensional unstationary convection–diffusion equation for instance, a conflicting situation between the source term time discretization and the boundary conditions may arise when using the standard Runge–Kutta method. We propose an alternative procedure by extending the Butcher Tableau to overcome this specific difficulty and achieve fourth-, sixth- or eighth-order of accuracy schemes in space and time. To this end, a new finite volume method is designed based on specific polynomial reconstructions for the space discretization, while we use the Extended Butcher Tableau to perform the time discretization. A large set of numerical tests has been carried out to validate the proposed method.

A construction of higher-order finite volume methods

A construction of higher-order finite volume methods

A new higher-order finite volume method based on Moving Least Squares for the resolution of the incompressible Navier–Stokes equations on unstructured grids

In this work a new higher-order (>2) accurate finite volume method for the resolution of the incompressible Navier–Stokes equations on unstructured grids is presented. The formulation is based on the use of Moving Least Squares (MLS) approximants. Third and fourth order accurate discretizations of the convective and viscous fluxes are obtained on collocated meshes. In addition, MLS is employed to design a new Momentum Interpolation Method that allows interpolations better than linear on any kind of mesh. The accuracy and performance of the proposed method is demonstrated by solving different steady and unsteady benchmark problems on unstructured grids.

Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

We propose a simple modification of standard WENO finite volume methods for Cartesian grids, which retains the full spatial order of accuracy of the one-dimensional discretization when applied to nonlinear multidimensional systems of conservation laws. We derive formulas, which allow us to compute high-order accurate point values of the conserved quantities at grid cell interfaces. Using those point values, we can compute a high-order flux at the center of a grid cell interface. Finally, we use those point values to compute high-order accurate averaged fluxes at cell interfaces as needed by a finite volume method. The method is described in detail for the two-dimensional Euler equations of gas dynamics. An extension to the three-dimensional case as well as to other nonlinear systems of conservation laws in divergence form is straightforward. Furthermore, similar ideas can be used to improve the accuracy of WENO type methods for hyperbolic systems which are not in divergence form. Several test computations confirm the high-order accuracy for smooth nonlinear problems.

Cell Conservative Flux Recovery and A Posteriori Error Estimate of Vertex-Centered Finite Volume Methods

AbstractA cell conservative flux recovery technique is developed here for vertex-centered finite volume methods of second order elliptic equations. It is based on solving a local Neumann problem on each control volume using mixed finite element methods. The recovered flux is used to construct a constant freea posteriorierror estimator which is proven to be reliable and efficient. Some numerical tests are presented to confirm the theoretical results. Our method works for general order finite volume methods and the recovery-based and residual-baseda posteriorierror estimators is the first result ona posteriorierror estimators for high order finite volume methods.

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.

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.

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.

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.

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.

Numerical simulations of high Lundquist number relativistic magnetic reconnection

We present the results of two-dimensional and three-dimensional magnetohydrodynamical numerical simulations of relativistic magnetic reconnection, with particular emphasis on the dynamics of the plasma in a Petschek-type configuration with high Lundquist numbers, S\sim 10^5-10^8. The numerical scheme adopted, allowing for unprecedented accuracy for this type of calculations, is based on high order finite volume and discontinuous Galerkin methods as recently proposed by \citet{Dumbser2009}. The possibility of producing high Lorentz factors is discussed, showing that Lorentz factors close to \sim 4 can be produced for a plasma parameter \sigma_m=20. Moreover, we find that the Sweet-Parker layers are unstable, generating secondary magnetic islands, but only for S > S_c = 10^8, much larger than what is reported in the Newtonian regime. Finally, the effects of a mildly anisotropic Ohm law are considered in a configuration with a guide magnetic field. Such effects produce only slightly faster reconnection rates and Lorentz factors of about 1% larger with respect to the perfectly isotropic Ohm law.

A study of dust grain screening in a weakly ionized plasma based on the numerical solution of the Vlasov-Bhatnagar-Gross-Krook kinetic equations

Screening of a dust grain in a weakly ionized plasma is studied for a wide range of collisional regimes. The problem is considered on the basis of the Vlasov-Bhatnagar-Gross-Krook equations for plasma particles. The equations are solved numerically on parallel processors by means of a high-order finite-volume method. The computations are carried out for different pressures of plasma background and different grain sizes. The values of the total grain charge, distributions of the electric potential, and basic macroscopic parameters of plasma particles (concentration, temperature, and velocity) are obtained. For two limiting cases of the collisionless and strongly collisional background, the obtained results are found to be in good agreement with those computed using the commonly known models (nonlinear collisionless models of Bystrenko et al. [Phys. Lett. A 299, 383 (2002)] and Ratynskaia et al. [Phys. Plasmas 13, 104508 (2006)] and drift-diffusion approach). For the transitional collisional regime, it is shown that the electric potential also has the Coulomb-type asymptotical behavior related to the presence of charging currents and existence of some residual (unscreened) grain charge. Both total and residual grain charges are found to increase as the collisional regime tends to the continuum limit. In addition, the influence of collisions on the ion distribution function near the grain surface is demonstrated.

Higher-order finite volume methods for elliptic boundary value problems

This paper studies higher-order finite volume methods for solving elliptic boundary value problems. We develop a general framework for construction and analysis of higher-order finite volume methods. Specifically, we establish the boundedness and uniform ellipticity of the bilinear forms for the methods, and show that they lead to an optimal error estimate of the methods. We prove that the uniform local-ellipticity of the family of the bilinear forms ensures its uniform ellipticity. We then establish necessary and sufficient conditions for the uniform local-ellipticity in terms of geometric requirements on the meshes of the domain of the differential equation, and provide a general way to investigate the mesh geometric requirements for arbitrary higher-order schemes. Several useful examples of higher-order finite volume methods are presented to illustrate the mesh geometric requirements.

High Order Finite Difference and Finite Volume Methods for Advection on the Sphere

Numerical schemes used for computational climate modeling and weather prediction are often of second order accuracy. It is well-known that methods of formal order higher than two offer a significan...

A high order moment method simulating evaporation and advection of a polydisperse liquid spray

In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gas–droplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N, N > 2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N-dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the Eulerian Multi Size Moment (EMSM) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor–Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.