High-order Finite Volume Method Research Articles

Three-dimensional numerical investigation on thermosolutal convection of power-law fluids in anisotropic porous media

The present study simulates the 3D unsteady double-diffusive natural convection subject to opposing thermal and solutal buoyancy forces (N<0) in a porous cubic by a generalized non-Darcy model, in which the effects of the crucial parameters such as the porous thermal Rayleigh numbers, buoyancy ratio and anisotropy ratio on the flow structure, heat and mass transfer of power-law fluids are investigated independently. The top and bottom walls are given different temperatures and concentrations, while the other walls are adiabatic and impermeable. A compact high order finite volume method is adopted to describe the flow structure and the resulting heat and mass transfer characteristic of non-Newtonian fluids in the anisotropy porous cubic. Our simulations show that the flow structure develops from conduction-dominated to convection-dominated as buoyancy ratio or anisotropy ratio or porous thermal Rayleigh number increases. The average Nusselt and Sherwood numbers keep constants during the conduction-dominated stage, then increase along the transition route. On the other hand, the impacts of different power-law indexes on the convection are mainly manifested in rheological properties, which elucidate that the shear-thinning fluids is more effective in heat and mass transfer enhancement than shear-thickening fluids. The studies may help us establish a physically reasonable methodology to systemically assess double-diffusive convection of non-Newtonian power-law fluids in anisotropy porous media in the real world.

A high-order finite-volume method for atmospheric flows on unstructured grids

A high-order finite-volume method for atmospheric flows on unstructured grids

An Essentially Non-Oscillatory Spectral Deferred Correction Method for Conservation Laws

We present a computational method based on the Spectral Deferred Corrections (SDC) time integration technique and the Essentially Non-Oscillatory (ENO) finite volume method for the conservation laws (one-dimensional Euler equations). The SDC technique is used to advance the solutions in time with high-order of accuracy. The ENO method is used to define high-order cell edge quantities that are then used to evaluate numerical fluxes. The coupling of the SDC method with a high-order finite volume method (Piece-wise Parabolic Method (PPM)) for solving the conservation laws is first carried out by Layton et al. in [Layton, A. T. and Minion, M. L. [2004] “Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics,” J. Comput. Phys. 194(2), 697–714]. Issues about this approach have been addressed and some improvements have been added to it in [Kadioglu et al. [2012] “A gas dynamics method based on the spectral deferred corrections (SDC) time integration technique and the piecewise parabolic method (PPM),” Am. J. Comput. Math. 1–4, 303–317]. Here, we investigate the implications when the PPM method is replaced with the well-known ENO method. We note that the SDC-PPM method is fourth-order accurate in time and space. Therefore, we kept the order of accuracy of the ENO procedure as fourth-order in order to be able to make a consistent comparison between the two approaches (SDC-ENO versus SDC-PPM methods). We have tested the new SDC-ENO technique by solving several test problems involving moderate to strong shock waves and smooth/complex flow structures. Our numerical results show that we have numerically achieved the formally fourth-order convergence of the new method for smooth problems. Our numerical results also indicate that the newly proposed technique performs very well providing highly resolved shock discontinuities and fairly good contact solutions. More importantly, the discontinuities in the flow test problems are captured with essentially no-oscillations. We have numerically compared the fourth-order SDC-ENO scheme to the fourth-order SDC-PPM method for the same test problems. The results are similar for most of the test problems except in some cases the SDC-PPM method suffers from minor oscillations compared to SDC-ENO scheme being completely oscillation free.

A high-order finite volume method on unstructured grids using RBF reconstruction

This paper proposes a high-order finite volume method based on radial basis function (RBF) reconstruction for the solution of Euler and Navier–Stokes equations on unstructured grids. Unlike traditional polynomial K-exact method, RBF method has stronger adaptability for different reconstruction stencils and more flexibility in choosing interpolating points. We expatiate on the detailed process of flow-field reconstruction by using multiquadric (MQ) basis function for the second-order and third-order schemes on unstructured triangular grids. Subsequently, we validate the accuracy order of RBF method through the numerical test case. Furthermore, the method is used to solve several typical flow fields. Compared with traditional K-exact high-order scheme, RBF method is more accurate and has lower numerical dissipation, which can obtain more elaborate and precise results.

Adjoint-based an adaptive finite volume method for steady Euler equations with non-oscillatory k -exact reconstruction

Abstract In [ An adaptive finite volume solver for steady Euler equations with non-oscillatory k-exact reconstruction, G. H. Hu and N. Y. Yi, J. Comput. Phys., Vol. 312, pp. 235–251 ], a high order finite volume method was proposed for steady Euler equations. The method consists of a Newton iteration method to linearize the governing equations, and a geometrical multigrid method to solve the derived system. The non-oscillatory k -exact reconstruction was developed towards the high quality solution reconstruction, and h -adaptive method was introduced to handle the efficiency issue. To further improve the numerical method, we study the following two issues in detail in this paper. First, we test the new reconstruction patch for the element which locates on the domain boundary, which could improve the convergence of the steady state significantly. Second, the adjoint-based a posteriori error estimation is introduced for the h -adaptive method, which could deliver more efficient adaptive refinement of the mesh. The numerical tests show the effectiveness of the proposed method.

Compact high order finite volume method on unstructured grids I: Basic formulations and one-dimensional schemes

The large reconstruction stencil has been the major bottleneck problem in developing high order finite volume schemes on unstructured grids. This paper presents a compact reconstruction procedure for arbitrarily high order finite volume method on unstructured grids to overcome this shortcoming. In this procedure, a set of constitutive relations are constructed by requiring the reconstruction polynomial and its derivatives on the control volume of interest to conserve their averages on face-neighboring cells. These relations result in an over-determined linear equation system, which, in the sense of least-squares, can be reduced to a block-tridiagonal system in the one-dimensional case. The one-dimensional formulations of the reconstruction are discussed in detail and a Fourier analysis is presented to study the dispersion/dissipation and stability properties. The WBAP limiter based on the secondary reconstruction is used to suppress the non-physical oscillations near discontinuities while achieve high order accuracy in smooth regions of the solution. Numerical results demonstrate the method's high order accuracy, robustness and shock capturing capability. A new reconstruction procedure for high order finite volume scheme is proposed.This procedure can achieve arbitrary order of accuracy using a compact stencil consisting of only face neighboring cells.The spectral properties of the proposed finite volume schemes are analyzed.Some test cases are solved numerically to show the superior performances of the proposed schemes.

An Essentially Nonoscillatory Spectral Deferred Correction Method for Hyperbolic Problems

We present a computational method based on the spectral deferred corrections (SDC) time integration technique and the essentially nonoscillatory (ENO) finite volume method for hyperbolic problems. The SDC technique is used to advance the solutions in time with high-order of accuracy. The ENO method is used to define high-order cell edge quantities that are then used to evaluate numerical fluxes. The coupling of the SDC method with a high-order finite volume method (piece-wise parabolic method (PPM)) is first carried out by Layton et al. [J. Comput. Phys. 194(2) (2004) 697]. Issues about this approach have been addressed and some improvements have been added to it in Kadioglu et al. [J. Comput. Math. 1(4) (2012) 303]. Here, we investigate the implications when the PPM method is replaced with the well-known ENO method. We note that the SDC-PPM method is fourth-order accurate in time and space. Therefore, we kept the order of accuracy of the ENO procedure as fourth-order in order to be able to make a consistent comparison between the two approaches (SDC-ENO versus SDC-PPM). We have tested the new SDC-ENO technique by solving smooth and nonsmooth hyperbolic problems. Our numerical results indicate that the fourth-order of accuracy in both space and time has been achieved for smooth problems. On the other hand, the new method performs very well when it is applied to nonlinear problems that involve discontinuous solutions. In other words, we have obtained highly resolved discontinuous solutions with essentially no-oscillations at or around the discontinuities.

Xtroem-fv: a new code for computational astrophysics based on very high order finite-volume methods – II. Relativistic hydro- and magnetohydrodynamics

xtroem-fv: a new code for computational astrophysics based on very high order finite-volume methods – II. Relativistic hydro- and magnetohydrodynamics

A new spectral finite volume method for elastic wave modelling on unstructured meshes

In this paper, we consider a new spectral finite volume method (FVM) for the elastic wave equations. Our new FVM is based on a piecewise constant approximation on a fine mesh and a high-order polynomial reconstruction on a coarser mesh. Our new method is constructed based on two existing techniques, the high-order FVM and the spectral FVM. In fact, we will construct a new method to take advantage of both methods. More precisely, our method has two distinctive features. The first one is that the local polynomial reconstructions are performed on the coarse triangles and the reconstruction matrices for all the coarse triangles are the same. This fact enhances the parallelization of our algorithm. We will present a parallel implementation of our method and show excellent efficiency results. The second one is that, by using a suitable number of finer triangles with a coarse triangle, we obtain an overdetermined reconstruction system, which can enhance the robustness of the reconstruction process. To derive our scheme, standard finite volume technique is applied to each fine triangle, and the high-order reconstructed polynomials, computed on coarse triangles, are used to compute numerical fluxes. We will present numerical results to show the performance of our method. Our method is presented for 2-D problems, but the same methodology can be applied to 3-D.

On spatial filtering of flow variables in high-order finite volume methods

A new method of spatial filtering in high-order finite volume methods is presented and assessed. The base of this method is to filter face-averaged variables (fluxes) and then the recovery of cell-averaged ones. Two kinds of filtering method are proposed. The first kind is highly dissipative and appropriate for the numerical regions that need high dissipation, e.g. sponge zones. The second kind, on the other hand, is a precise method and hence is suitable for applying the high-order finite difference filters to the finite volume methods. Applying high-order finite difference filters directly to the high-order finite volume methods without using the proposed method causes stability problems in the large eddy simulation of high Reynolds number flows. The test cases, namely, propagation of the one-dimensional wave packet, advection of a two-dimensional vortical wave and the large eddy simulation of turbulent round jet at Re=104 and 105 are used to examine the accuracy and performance of the proposed methods. The last test case shows the effectiveness of the proposed methods in the large eddy simulation of flows by high-order FVMs. The performance of filtering methods on a curvilinear wavy grid is also investigated.

Divergence-Free WENO Reconstruction-Based Finite Volume Scheme for Solving Ideal MHD Equations on Triangular Meshes

AbstractIn this paper, we introduce a high-order accurate constrained transport type finite volume method to solve ideal magnetohydrodynamic equations on two-dimensional triangular meshes. A new divergence-free WENO-based reconstruction method is developed to maintain exactly divergence-free evolution of the numerical magnetic field. In this formulation, the normal component of the magnetic field at each face of a triangle is reconstructed uniquely and with the desired order of accuracy. Additionally, a new weighted flux interpolation approach is also developed to compute the z-component of the electric field at vertices of grid cells. We also present numerical examples to demonstrate the accuracy and robustness of the proposed scheme.

Compact high order finite volume method on unstructured grids II: Extension to two-dimensional Euler equations

Abstract In this paper, the compact least-squares finite volume method on unstructured grids proposed in our previous paper is extended to multi-dimensional systems, namely the two-dimensional Euler equations. The key element of this scheme is the compact least-squares reconstruction in which a set of constitutive relations are constructed by requiring the reconstruction polynomial and its spatial derivatives on the control volume of interest to conserve their averages on the face-neighboring cells. These relations result in an over-determined linear equation system. A large sparse system of linear equations is resulted by using the least-squares technique. An efficient solution strategy is of crucial importance for the application of the proposed scheme in multi-dimensional problems since both direct and iterative solvers for this system are computationally very expensive. In the present paper, it is found that in the cases of steady flow simulation and unsteady flow simulation using dual time stepping technique, the present reconstruction method can be coupled with temporal discretization scheme to achieve high computational efficiency. The WBAP limiter and a problem-independent shock detector are used in the simulation of flow with discontinuities. Numerical results demonstrate the high order accuracy, high computational efficiency and capability of handling both complex physics and geometries of the proposed schemes.

High-order finite-volume methods on locally-structured grids

We present an approach to designing arbitrarily high-order finite-volume spatial discretizations on locally-rectangular grids. It is based on the use of a simple class of high-order quadratures for computing the average of fluxes over faces. This approach has the advantage of being a variation on widely-used second-order methods, so that the prior experience in engineering those methods carries over in the higher-order case. Among the issues discussed are the basic design principles for uniform grids, the extension to locally-refined nest grid hierarchies, and the treatment of complex geometries using mapped grids, multiblock grids, and cut-cell representations.

A cell-centered multipoint flux approximation method with a diamond stencil coupled with a higher order finite volume method for the simulation of oil–water displacements in heterogeneous and anisotropic petroleum reservoirs

The simulation of fluid flow and transport in heterogeneous and anisotropic oil and gas reservoirs poses a great challenge from the numerical point of view, due to the modeling of complex depositional environments, including inclined laminated layers, channels, fractures and faults and the modeling of deviated wells, making it difficult to build and handle the Reservoir Characterization Process (RCM), particularly by using structured meshes (cartesian or corner point), which is the current pattern in petroleum reservoir simulators. Under certain hypotheses, the mathematical model that describes the fluid flow in petroleum reservoirs includes an elliptic equation with heterogeneous, possibly discontinuous, coefficients for the pressure field and a non-linear hyperbolic equation for the saturation field. In the present paper, these equations are solved via an Implicit Pressure Explicit Saturation (IMPES) procedure. To solve these equations, we use a full cell-centered finite volume formulation. The pressure equation is discretized by a non-orthodox Multipoint Flux Approximation Method with a Diamond type stencil (MPFA-D) and used for the first time for the solution of two-phase flow problems in heterogeneous porous media. It is very robust and capable of reproducing piecewise linear solutions exactly by means of a linear preserving interpolation with explicit weights that avoids the solution of locally defined systems of equations. For the solution of the saturation equation, we use a Monotone Upstream Centered Scheme for Conservation Laws (MUSCL) method based on a gradient reconstruction obtained by a least square technique in which monotonicity is reinforced by an appropriate slope limiter. The method can be used with general polygonal meshes, even though we restrict ourselves to conforming triangular and quadrilateral grids. In order to validate and show the robustness of our formulation, we solve some problems including heterogeneous and anisotropic reservoirs and displacements with high mobility ratios. Our results compare well with others found in literature.

A high order finite volume method for one dimensional nonlocal reactive flows of parabolic type

In this paper, we consider a high order finite volume approximation of one‐dimensional nonlocal reactive flows of parabolic type. The method is obtained by discretizing in space by arbitrary order vertex‐centered finite volumes, followed by a modified Simpson quadrature scheme for the time stepping. Compared to the existed finite volume methods, this new finite volume scheme could achieve the desired accuracy with less data storage by employing higher‐order trial spaces. The finite volume approximations are proved to possess optimal order convergence rates in the H1‐norm and L2‐norm, which are also confirmed by numerical tests. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Xtroem-fv: a new code for computational astrophysics based on very high order finite-volume methods – I. Magnetohydrodynamics

xtroem-fv: a new code for computational astrophysics based on very high order finite-volume methods – I. Magnetohydrodynamics

A high-order density-based finite volume method for the computation of all-speed flows

In this paper we present a high-order density-based finite-volume framework for all-speed flows. The formulation is based on high-order variable reconstructions performed using Moving Least Squares approximations. In particular, we show that combining high-order discretization schemes with low-Mach fixes, it is possible to remove the grid dependency problem at low Mach numbers on both structured and unstructured grids. In order to maintain the accuracy and the robustness of the numerical method at transonic conditions, different procedures are proposed, based on the use of a selective limiting.

A High-Order Piecewise Polynomial Reconstruction for Finite-Volume Methods Solving Convection and Diffusion Equations

In finite-volume methods for fluid flows, the average of field variables over local mesh cells are the unknowns that are integrated in time based on the integral conservation laws. In order to compute the cell-face fluxes, nodal variables and derivative values at cell faces are needed, which ultimately determine the accuracy of the finite-volume method. In this work, a piecewise fourth-order polynomial reconstruction model based on volume averages is developed for smoothly varying flow fields over finite-volume cells. The obtained polynomial is used to extrapolate the cell-face variables and derivative values with high-order accuracy. The extrapolated cell-face quantities are used directly to compute the convection or diffusion integrals to construct high-order finite-volume methods. Some one-dimensional examples are shown to demonstrate the fifth-order accuracy of the proposed approach when solving the linear convection equation, and the fourth-order solution accuracy when solving the linear diffusion equation.

High Order Maximum Principle Preserving Finite Volume Method for Convection Dominated Problems

In this paper, we investigate the application of the maximum principle preserving (MPP) parametrized flux limiters to the high order finite volume scheme with Runge-Kutta time discretization for solving convection dominated problems. Such flux limiter was originally proposed in [Xu, Math. Comp., 2013] and further developed in [Xiong et. al., J. Comp. Phys., 2013] for finite difference WENO schemes with Runge-Kutta time discretization for convection equations. The main idea is to limit the temporal integrated high order numerical flux toward a first order MPP monotone flux. In this paper, we generalize such flux limiter to high order finite volume methods solving convection-dominated problems, which is easy to implement and introduces little computational overhead. More importantly, for the first time in the finite volume setting, we provide a general proof that the proposed flux limiter maintains high order accuracy of the original WENO scheme for linear advection problems without any additional time step restriction. For general nonlinear convection-dominated problems, we prove that the proposed flux limiter introduces up to $O(\Delta x^3+\Delta t^3)$ modification to the high order temporal integrated flux in the original WENO scheme without extra time step constraint. We also numerically investigate the preservation of up to ninth order accuracy of the proposed flux limiter in a general setting. The advantage of the proposed method is demonstrated through various numerical experiments.

An adaptive multiblock high-order finite-volume method for solving the shallow-water equations on the sphere

We present a high-order finite-volume approach for solving the shallow-water equations on the sphere, using multiblock grids on the cubed-sphere. This approach combines a Runge--Kutta time discretization with a fourth-order accurate spatial discretization, and includes adaptive mesh refinement and refinement in time. Results of tests show fourth-order convergence for the shallow-water equations as well as for advection in a highly deformational flow. Hierarchical adaptive mesh refinement allows solution error to be achieved that is comparable to that obtained with uniform resolution of the most refined level of the hierarchy, but with many fewer operations.