High-order Finite Volume Research Articles

A numerical scheme for an improved Green–Naghdi model in the Camassa–Holm regime for the propagation of internal waves

In this paper we introduce a new reformulation of the Green–Naghdi model in the Camassa–Holm regime for the propagation of internal waves over a flat topography derived by Duchêne et al. [18]. These new Green–Naghdi systems are adapted to improve the frequency dispersion of the original model, they share the same order of precision as the standard one but have an appropriate structure which makes them much more suitable for the numerical resolution. We develop a second order splitting scheme where the hyperbolic part of the system is treated with a high-order finite volume scheme and the dispersive part is treated with a finite difference approach. Numerical simulations are then performed to validate the model and the numerical methods.

A high-order finite volume method for solving one-dimensional convection and diffusion equations

ABSTRACTA spatially high-order finite volume method for solving convection and diffusion equations is developed and tested in this work. The method performs a high-order piecewise polynomial reconstruction of the local flow field based on the relationship between Taylor’s series expansion and the volume-averaged flow quantities. A 5 × 5 matrix inversion for each cell is done to compute the cell-center variables and derivatives up to fourth order. While a fixed symmetric grid stencil is maintained in smooth flow regions, a detector for large change in linear data slopes is developed to trigger the use of ENO stencil around flow discontinuities. Regular time integration scheme such as the four-stage Runge–Kutta method or the Euler implicit method is used for time integration. The present finite volume method is shown to be spatially fifth-order accurate for the linear convection equation, fourth-order accurate for the linear diffusion equation, and fourth-order accurate for the linear convection–diffusion equation. The shocks captured in solving the inviscid Burger’s equation are sharp and oscillation free. For the system of Euler equations, a characteristic limiter is further developed to limit the growth of total variation of the solution. Test examples solving shock-tube problems and the interactions of two blast waves show that various flow discontinuities are captured sharply without spurious oscillations.

Relativistic Vlasov-Maxwell modelling using finite volumes and adaptive mesh refinement

The dynamics of collisionless plasmas can be modelled by the Vlasov-Maxwell system of equations. An Eulerian approach is needed to accurately describe processes that are governed by high energy tails in the distribution function, but is of limited efficiency for high dimensional problems. The use of an adaptive mesh can reduce the scaling of the computational cost with the dimension of the problem. Here, we present a relativistic Eulerian Vlasov-Maxwell solver with block-structured adaptive mesh refinement in one spatial and one momentum dimension. The discretization of the Vlasov equation is based on a high-order finite volume method. A flux corrected transport algorithm is applied to limit spurious oscillations and ensure the physical character of the distribution function. We demonstrate a speed-up by a factor of 7 × in a typical scenario involving laser pulse interaction with an underdense plasma due to the use of an adaptive mesh.Graphical abstract

Admissible Regions for Higher-Order Finite Volume Method Grids

AbstractAdmissible regions for higher-order finite volume method (FVM) grids are considered. A new Hermite quintic FVM and a new hybrid quintic FVM are constructed to solve elliptic boundary value problems, and the corresponding admissible regions are investigated. A sufficient condition for the uniform local-ellipticity of the new hybrid quintic FVM is obtained when its admissible region is known. In addition, the admissible regions for a large number of higher-order FVMs are provided. For the same class of FVM (Lagrange, Hermite or hybrid), the higher order FVM has a smaller admissible region such that stronger geometric restrictions are required to guarantee its uniform local-ellipticity.

The Piecewise Cubic Method (PCM) for computational fluid dynamics

We present a new high-order finite volume reconstruction method for hyperbolic conservation laws. The method is based on a piecewise cubic polynomial which provides its solutions a fifth-order accuracy in space. The spatially reconstructed solutions are evolved in time with a fourth-order accuracy by tracing the characteristics of the cubic polynomials. As a result, our temporal update scheme provides a significantly simpler and computationally more efficient approach in achieving fourth order accuracy in time, relative to the comparable fourth-order Runge–Kutta method. We demonstrate that the solutions of PCM converges at fifth-order in solving 1D smooth flows described by hyperbolic conservation laws. We test the new scheme on a range of numerical experiments, including both gas dynamics and magnetohydrodynamics applications in multiple spatial dimensions.

A new high-order finite volume method for 3D elastic wave simulation on unstructured meshes

In this paper, we proposed a new efficient high-order finite volume method for 3D elastic wave simulation on unstructured tetrahedral meshes. With the relative coarse tetrahedral meshes, we make subdivision in each tetrahedron to generate a stencil for the high-order polynomial reconstruction. The subdivision algorithm guarantees the number of subelements is greater than the degrees of freedom of a complete polynomial. We perform the reconstruction on this stencil by using cell-averaged quantities based on the hierarchical orthonormal basis functions. Unlike the traditional high-order finite volume method, our new method has a very local property like DG and can be written as an inner-split computational scheme which is beneficial to reducing computational amount. Moreover, the stencil in our method is easy to generate for all tetrahedrons especially in the three-dimensional case. The resulting reconstruction matrix is invertible and remains unchanged for all tetrahedrons and thus it can be pre-computed and stored before time evolution. These special advantages facilitate the parallelization and high-order computations. We show convergence results obtained with the proposed method up to fifth order accuracy in space. The high-order accuracy in time is obtained by the Runge–Kutta method. Comparisons between numerical and analytic solutions show the proposed method can provide accurate wavefield information. Numerical simulation for a realistic model with complex topography demonstrates the effectiveness and potential applications of our method. Though the method is proposed based on the 3D elastic wave equation, it can be extended to other linear hyperbolic system.

Accuracy preserving limiter for the high-order finite volume method on unstructured grids

This paper proposes a distance weighted biased averaging procedure (DWBAP) and applies it to the high-order finite volume method on unstructured grids. Unlike the initial BAP limiter, we use the inverse distance weighting for biased function of the derivatives of flow variables, and meanwhile introduce the standard deviation coefficient of the reconstruction stencil to determine the value of the distance weighting. This can efficiently maintain both the property of high precision in smooth regions and control numerical oscillations in discontinuities. Through various numerical examples, it is demonstrated that the developed DWBAP limiter can capture strong shock waves and have a good property of convergence.

A Simple Bound-Preserving Sweeping Technique for Conservative Numerical Approximations

In this paper, we propose a simple bound-preserving sweeping procedure for conservative numerical approximations. Conservative schemes are of importance in many applications, yet for high order methods, the numerical solutions do not necessarily satisfy maximum principle. This paper constructs a simple sweeping algorithm to enforce the bound of the solutions. It has a very general framework acting as a postprocessing step accommodating many point-based or cell average-based discretizations. The method is proven to preserve the bounds of the numerical solution while conserving a prescribed quantity designated as a weighted average of values from all points. The technique is demonstrated to work well with a spectral method, high order finite difference and finite volume methods for scalar conservation laws and incompressible flows. Extensive numerical tests in 1D and 2D are provided to verify the accuracy of the sweeping procedure.

A Non-oscillatory Multi-Moment Finite Volume Scheme with Boundary Gradient Switching

In this work we propose a new formulation for high-order multi-moment constrained finite volume (MCV) method. In the one-dimensional building-block scheme, three local degrees of freedom (DOFs) are equidistantly defined within a grid cell. Two candidate polynomials for spatial reconstruction of third-order are built by adopting one additional constraint condition from the adjacent cells, i.e. the DOF at middle point of left or right neighbour. A boundary gradient switching (BGS) algorithm based on the variation-minimization principle is devised to determine the spatial reconstruction from the two candidates, so as to remove the spurious oscillations around the discontinuities. The resulted non-oscillatory MCV3-BGS scheme is of fourth-order accuracy and completely free of case-dependent ad hoc parameters. The widely used benchmark tests of one- and two-dimensional scalar and Euler hyperbolic conservation laws are solved to verify the performance of the proposed scheme in this paper. The MCV3-BGS scheme is very promising for the practical applications due to its accuracy, non-oscillatory feature and algorithmic simplicity.

Compact high order finite volume method on unstructured grids III: Variational reconstruction

This paper presents a variational reconstruction for the high order finite volume method in solving the two-dimensional Navier–Stokes equations on arbitrary unstructured grids. In the variational reconstruction, an interfacial jump integration is defined to measure the jumps of the reconstruction polynomial and its spatial derivatives on each cell interface. The system of linear equations to determine the reconstruction polynomials is derived by minimizing the total interfacial jump integration in the computational domain using the variational method. On each control volume, the derived equations are implicit relations between the coefficients of the reconstruction polynomials defined on a compact stencil involving only the current cell and its direct face-neighbors. The reconstruction and time integration coupled iteration method proposed in our previous paper is used to achieve high computational efficiency. A problem-independent shock detector and the WBAP limiter are used to suppress non-physical oscillations in the simulation of flow with discontinuities. The advantages of the finite volume method using the variational reconstruction over the compact least-squares finite volume method proposed in our previous papers are higher accuracy, higher computational efficiency, more flexible boundary treatment and non-singularity of the reconstruction matrix. A number of numerical test cases are solved to verify the accuracy, efficiency and shock-capturing capability of the finite volume method using the variational reconstruction.

A Higher-Order Chimera Method for Finite Volume Schemes

In this work a higher-order accurate finite volume method for the resolution of the Euler/Navier–Stokes equations using Chimera grid techniques is presented. The formulation is based on the use of Moving Least Squares approximations in order to obtain higher-order accurate reconstruction and connectivity between the overlapped grids. The accuracy and performance of the proposed methodology is demonstrated by solving different benchmark problems.

Error estimates for higher-order finite volume schemes for convection–diffusion problems

AbstractIt is still an open problem to provea priorierror estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domainΩin ℝ2and we can prove such kind of ana priorierror estimate. In the part of the estimate, which refers to the discretization of the convective term, we gainh1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.

Accurate simulation of transient landscape evolution by eliminating numerical diffusion: the TTLEM 1.0 model

Abstract. Landscape evolution models (LEMs) allow the study of earth surface responses to changing climatic and tectonic forcings. While much effort has been devoted to the development of LEMs that simulate a wide range of processes, the numerical accuracy of these models has received less attention. Most LEMs use first-order accurate numerical methods that suffer from substantial numerical diffusion. Numerical diffusion particularly affects the solution of the advection equation and thus the simulation of retreating landforms such as cliffs and river knickpoints. This has potential consequences for the integrated response of the simulated landscape. Here we test a higher-order flux-limiting finite volume method that is total variation diminishing (TVD-FVM) to solve the partial differential equations of river incision and tectonic displacement. We show that using the TVD-FVM to simulate river incision significantly influences the evolution of simulated landscapes and the spatial and temporal variability of catchment-wide erosion rates. Furthermore, a two-dimensional TVD-FVM accurately simulates the evolution of landscapes affected by lateral tectonic displacement, a process whose simulation was hitherto largely limited to LEMs with flexible spatial discretization. We implement the scheme in TTLEM (TopoToolbox Landscape Evolution Model), a spatially explicit, raster-based LEM for the study of fluvially eroding landscapes in TopoToolbox 2.

Assessment of high-order finite volume methods on unstructured meshes for RANS solutions of aeronautical configurations

This paper is concerned with the application of k-exact finite volume methods for compressible Reynolds-Averaged Navier-Stokes computations of flows around aeronautical configurations including the NACA0012, RAE2822, MDA30P30N, ONERA-M6, CRM and DLR-F11. High-order spatial discretisation is obtained with the Weighted Essentially Non-Oscillatory and the Monotone-Upstream Central Scheme for Conservation Laws methods on hybrid unstructured grids in two- and three- dimensions. Schemes of fifth, third and second order comprise the foundation of the analysis, with main findings suggesting that enhanced accuracy can be obtained with at least a third-order scheme. Steady state solutions are achieved with the implicit approximately factored Lower-Upper Symmetric Gauss-Seidel time advancing technique, convergence properties of each scheme are discussed. The Spalart-Allmaras turbulence model is employed where its discretisation with respect to the high-order framework is assessed. A low-Mach number treatment technique is studied, where recovery of accuracy in low speed regions is exemplified. Results are compared with referenced data and discussed in terms of accuracy, grid dependence and computational budget.

Central Weighted ENO Schemes for Hyperbolic Conservation Laws on Fixed and Moving Unstructured Meshes

We present a novel arbitrary high order accurate central WENO spatial reconstruction procedure (CWENO) for the solution of nonlinear systems of hyperbolic conservation laws on fixed and moving unstructured simplex meshes in two and three space dimensions. Starting from the given cell averages of a function on a triangular or tetrahedral control volume and its neighbors, the nonlinear CWENO reconstruction yields a high order accurate and essentially non-oscillatory polynomial that is defined everywhere in the cell. Compared to other WENO schemes on unstructured meshes, the total stencil size is the minimum possible one, as in classical point-wise WENO schemes of Jiang and Shu. However, the linear weights can be chosen arbitrarily, which makes the practical implementation on general unstructured meshes particularly simple. We make use of the piecewise polynomials generated by the CWENO reconstruction operator inside the framework of fully discrete and high order accurate one-step ADER finite volume schemes on fixed Eulerian grids as well as on moving Arbitrary-Lagrangian-Eulerian (ALE) meshes. The computational efficiency of the high order finite volume schemes based on the new CWENO reconstruction is tested on several two- and three-dimensional systems of hyperbolic conservation laws and is found to be more efficient in terms of memory consumption and computational efficiency with respect to classical WENO reconstruction schemes on unstructured meshes. We also provide evidence that the new algorithm is suitable for implementation on massively parallel distributed memory supercomputers, showing two numerical examples run with more than one billion degrees of freedom in space. To our knowledge, at present these are the largest simulations ever run with unstructured WENO finite volume schemes.

Higher-order unstructured finite volume RANS solution of turbulent compressible flows

Abstract In this paper, we describe a higher-order finite volume solver for the RANS solution of turbulent compressible flows on unstructured meshes. Considering that higher-order least-squares reconstruction suffers from poor accuracy on highly anisotropic meshes with curvature, we propose a modification to this method which highly improves the accuracy of solution approximation and output quantities. In addition, robust implementation and coupling of a RANS turbulence model with a higher-order finite volume solver is described and an efficient solution strategy is adopted for the convergence to the steady-state solution. We present our results, ranging from second- to fourth-order, for the test case of turbulent flat plate, subsonic and transonic flows over two-dimensional airfoils and also turbulent flow over a multi-element configuration. These results are verified by comparison against well-established numerical and experimental values in the literature. Our results show the advantages of higher-order methods in obtaining a more accurate solution with fewer degrees of freedom and also fast and efficient convergence to the steady-state solutions.

Entropy generation due to three-dimensional double-diffusive convection of power-law fluids in heterogeneous porous media

The article reports a numerical study of entropy generation due to the double-diffusive natural convection in a 3D heterogeneous porous cubic saturated with power-law fluids and submitted to horizontal thermal and concentration gradients. The governing equations based on a generalized non-Darcy model are solved by using the compact high order finite volume method. This approach is devised to detect the effects of the heterogeneity level, the buoyancy ratio and the porous thermal Rayleigh number on the fluid flow and its associated irreversibility characteristics. The results show that the mean Nusselt and Sherwood number reduce, while the total entropy generation increases, as the level of heterogeneity increases by the exponential distribution of the permeability; the mean Nusselt, Sherwood number and total entropy generation increases as the porous thermal Rayleigh number increases or the buoyancy ratio increases (decreases) for the thermal dominated flow (solutal dominated flow) by the heat and mass transfer performance. Apart from that, our numerical tests of the approach on the shear-thinning, the Newtonian and shear-thickening fluids show that the impacts of different power-law indexes on the entropy generation due to fluid friction, heat and mass transfer are mainly manifested in rheological properties, which elucidate that the shear-thinning fluids is more effective than shear-thickening fluids. The studies may help us establish a physically reasonable methodology to systemically assess fluid flow and energy consumption in heterogeneous porous media in the real world.

High order numerical methods for networks of hyperbolic conservation laws coupled with ODEs and lumped parameter models

In this paper we construct high order finite volume schemes on networks of hyperbolic conservation laws with coupling conditions involving ODEs. We consider two generalized Riemann solvers at the junction, one of Toro–Castro type and a solver of Harten, Enquist, Osher, Chakravarthy type. The ODE is treated with a Taylor method or an explicit Runge–Kutta scheme, respectively. Both resulting high order methods conserve quantities exactly if the conservation is part of the coupling conditions. Furthermore we present a technique to incorporate lumped parameter models, which arise from simplifying parts of a network. The high order convergence and the robust capturing of shocks are investigated numerically in several test cases.

Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations

Fisher and Carpenter (2013) [12] found a remarkable equivalence of general diagonal norm high-order summation-by-parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux. We show that besides the construction of entropy stable high-order schemes, a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms. Thus, by changing the subcell finite volume flux to a specific choice, we are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms, such as the Ducros splitting and the Kennedy and Gruber splitting. Although these split forms are not entropy stable, we present a systematic way to prove which of those split forms are at least kinetic energy preserving. With this, we construct a unified high-order split form DG framework. We investigate with three dimensional numerical simulations of the inviscid Taylor–Green vortex and show that the new split forms enhance the robustness of high-order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows. In fact, we show that for certain test cases, the novel split form discontinuous Galerkin schemes are more robust than the discontinuous Galerkin scheme with over-integration.

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.