High-order Finite Volume Research Articles

A new class of fully nonlinear and weakly dispersive Green–Naghdi models for efficient 2D simulations

We introduce a new class of two-dimensional fully nonlinear and weakly dispersive Green–Naghdi equations over varying topography. These new Green–Naghdi systems share the same order of precision as the standard one but have a mathematical structure which makes them much more suitable for the numerical resolution, in particular in the demanding case of two dimensional surfaces.For these new models, we develop a high order, well balanced, and robust numerical code relying on a hybrid finite volume and finite difference splitting approach. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a finite difference approach. Higher order accuracy in space and time is achieved through WENO reconstruction methods and through an SSP-RK time stepping. Particular effort is made to ensure positivity of the water depth.Numerical validations are then performed, involving one and two dimensional cases and showing the ability of the resulting numerical model to handle waves propagation and transformation, wetting and drying; some simple treatments of wave breaking are also included. The resulting numerical code is particularly efficient from a computational point of view and very robust; it can therefore be used to handle complex two dimensional configurations.

A Maximum-Principle-Satisfying High-Order Finite Volume Compact WENO Scheme for Scalar Conservation Laws with Applications in Incompressible Flows

In this paper, a maximum-principle-satisfying finite volume compact scheme is proposed for solving scalar hyperbolic conservation laws. The scheme combines weighted essentially non-oscillatory schemes (WENO) with a class of compact schemes under a finite volume framework, in which the nonlinear WENO weights are coupled with lower order compact stencils. The maximum-principle-satisfying polynomial rescaling limiter in Zhang and Shu (J Comput Phys 229:3091---3120, 2010, Proc R Soc A Math Phys Eng Sci 467:2752---2776, 2011) is adopted to construct the present schemes at each stage of an explicit Runge---Kutta method, without destroying high order accuracy and conservativity. Numerical examples for one and two dimensional problems including incompressible flows are presented to assess the good performance, maximum principle preserving, essentially non-oscillatory and high resolution of the proposed method.

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.

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 New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

AbstractIn this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions witha posterioridetection and polynomial degree decrementing processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.

A construction of higher-order finite volume methods

A construction of higher-order finite volume methods

A Padé compact high-order finite volume scheme for nonlinear Schrödinger equations

In this work, a Padé compact high-order finite volume scheme is presented for the solution of one-dimensional nonlinear Schrödinger equations. The compact high-order finite volume schemes posses inherent conservation of the equations and high order accuracy within small stencils. Fourier error analysis demonstrates that the spectral resolution of the Padé compact finite volume scheme exceeds that of the standard finite volume schemes in terms of the same order of accuracy. Besides, the linear stability of the temporal discretization scheme is also performed by using the Fourier analysis. Numerical results are obtained for the nonlinear Schrödinger equations with various initial and boundary conditions, which manifests high accuracy and validity of the Padé compact finite volume scheme.

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.

ADER schemes and high order coupling on networks of hyperbolic conservation laws

In this article we present a method to extend high order finite volume schemes to networks of hyperbolic conservation laws with algebraic coupling conditions. This method is based on an ADER approach in time to solve the generalized Riemann problem at the junction. Additionally to the high order accuracy, this approach maintains an exact conservation of quantities if stated by the coupling conditions. Several numerical examples confirm the benefits of a high order coupling procedure for high order accuracy and stable shock capturing.

A higher order Finite Volume resolution method for a system related to the inviscid primitive equations in a complex domain

We construct the cell-centered Finite Volume discretization of the two-dimensional inviscid primitive equations in a domain with topography. To compute the numerical fluxes, the so-called Upwind Scheme (US) and the Central-Upwind Scheme (CUS) are introduced. For the time discretization, we use the classical fourth order Runge---Kutta method. We verify, with our numerical simulations, that the US (or CUS) is a robust first (or second) order scheme, regardless of the shape or size of the topography and without any mesh refinement near the topography.

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.

Large Eddy Simulation of Pseudo-Shock Waves Using Wall Model

A supersonic turbulent flowfield involving the pseudo-shock waves in an isolator of a supersonic combustion ramjet is computed using two different LES codes which are a high-order upwind finite volume scheme, and a sixth order compact differencing scheme utilizing the localized artificial diffusivity method for stabilizing shock waves and employing a wall model to enable the use of coarse mesh. In the validation study where a supersonic turbulent boundary layer flow over a flat plate is examined, both LES codes are well validated using velocity profile in the boundary layer given by the hot-wire anemometry and normal stress given by the laser Doppler anemometry. In particular, the sixth order compact differencing scheme gives closer agreements with these experimental data. Then, the validated LES codes are applied to solve the Mach number 2.5 supersonic turbulent flowfield involving the pseudo-shock waves. It is shown that typical features of unsteady flowfield of the pseudo-shock waves are well obtained by both schemes. Again, it is indicated that the sixth order compact differencing scheme gives closer agreements with the existing velocity data obtained by particle image velocimetry and pressure fluctuation data on the wall surface. Besides, the computational cost of the compact differencing scheme is found to be 1/7 of that for the upwind finite volume scheme, even though a wall model is solved at each grid point on the wall surface. Therefore, the obtained results in the present study allow recommending the sixth order compact differencing scheme with a wall model for simulating supersonic turbulent flowfield in an isolator involving the pseudo-shock waves.

Performance comparison of flux schemes for numerical simulation of high-speed inviscid flows

Numerical investigations to assess the performance of different flux schemes for the spatial discretisation of the Euler equations have been performed. The schemes employed in the study include flux vector and flux difference splitting schemes as well as hybrid schemes. The schemes are cast in an unstructured high-order finite volume framework and are studied on typical engineering problems in the supersonic and hypersonic regimes. While all the flux schemes perform well in the supersonic and low hypersonic range, their performance show a marked change in the high Mach number regimes. Numerical experiments suggest that the AUSM family of schemes is the most accurate while the Rusanov scheme is the most robust for the range of Mach numbers considered in the study. These studies indicate that a robust and accurate numerical solver for high-speed compressible flows necessitate the use of blended schemes that provide a right balance between accuracy and numerical dissipation.

Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem

In this paper, we present a class of parametrized limiters used to achieve strict maximum principle for high order numerical schemes applied to hyperbolic conservation laws computation. By decoupling a sequence of parameters embedded in a group of explicit inequalities, the numerical fluxes are locally redefined in consistent and conservative formulation. We will show that the global maximum principle can be preserved while the high order accuracy of the underlying scheme is maintained. The parametrized limiters are less restrictive on the CFL number when applied to high order finite volume scheme. The less restrictive limiters allow for the development of the high order finite difference scheme which preserves the maximum principle. Within the proposed parametrized limiters framework, a successive sequence of limiters are designed to allow for significantly large CFL number by relaxing the limits on the intermediate values of the multistage Runge-Kutta method. Numerical results and preliminary analysis for linear and nonlinear scalar problems are presented to support the claim. The parametrized limiters are applied to the numerical fluxes directly. There is no increased complexity to apply the parametrized limiters to different kinds of monotone numerical fluxes.

Well-balanced schemes for the Euler equations with gravitation

Well-balanced high-order finite volume schemes are designed to approximate the Euler equations with gravitation. The schemes preserve discrete equilibria, corresponding to a large class of physically stable hydrostatic steady states. Based on a novel local hydrostatic reconstruction, the derived schemes are well-balanced for any consistent numerical flux for the Euler equations. The form of the hydrostatic reconstruction is both very simple and computationally efficient as it requires no analytical or numerical integration. Moreover, as required by many interesting astrophysical scenarios, the schemes are applicable beyond the ideal gas law. Both first- and second-order accurate versions of the schemes and their extension to multi-dimensional equilibria are presented. Several numerical experiments demonstrating the superior performance of the well-balanced schemes, as compared to standard finite volume schemes, are also presented.

An integral contravariant formulation of the fully non-linear Boussinesq equations

In this paper we propose an integral form of the fully non-linear Boussinesq equations in contravariant formulation, in which Christoffel symbols are avoided, in order to simulate wave transformation phenomena, wave breaking and nearshore currents in computational domains representing the complex morphology of real coastal regions. Following the approach proposed by Chen (2006), the motion equations retain the term related to the approximation to the second order of the vertical vorticity. A new Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the fully non-linear Boussinesq equations on generalised curvilinear coordinate systems is proposed. The equations are rearranged in order to solve them by a high resolution hybrid finite volume–finite difference scheme. The conservative part of the above-mentioned equations, consisting of the convective terms and the terms related to the free surface elevation, is discretised by a high-order shock-capturing finite volume scheme in which an exact Riemann solver is involved; dispersive terms and the term related to the approximation to the second order of the vertical vorticity are discretised by a cell-centred finite difference scheme. The shock-capturing method makes it possible to intrinsically model the wave breaking, therefore no additional terms are needed to take into account the breaking related energy dissipation in the surf zone. The model is verified against several benchmark tests, and the results are compared with experimental, theoretical and alternative numerical solutions.

High order space–time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems

We present a class of high order finite volume schemes for the solution of non-conservative hyperbolic systems that combines the one-step ADER-WENO finite volume approach with space–time adaptive mesh refinement (AMR). The resulting algorithm, which is particularly well suited for the treatment of material interfaces in compressible multi-phase flows, is based on: (i) high order of accuracy in space obtained through WENO reconstruction, (ii) a high order one-step time discretization via a local space–time discontinuous Galerkin predictor method, and (iii) the use of a path conservative scheme for handling the non-conservative terms of the equations. The AMR property with time accurate local time stepping, which has been treated according to a cell-by-cell strategy, strongly relies on the high order one-step time discretization, which naturally allows a high order accurate and consistent computation of the jump terms at interfaces between elements using different time steps. The new scheme has been successfully validated on some test problems for the Baer–Nunziato model of compressible multiphase flows.

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.

High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: Applications to compressible multi-phase flows

In this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a non-linear WENO reconstruction operator on unstructured triangular meshes. High order accuracy in time is obtained via a local Lagrangian space–time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space–time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian–Eulerian (ALE) method, where the mesh velocity can be chosen independently of the fluid velocity.The new scheme is applied to the full seven-equation Baer–Nunziato model of compressible multi-phase flows with relaxation source terms in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer–Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided.

Adaptation and optimization of basic operations for an unstructured mesh CFD algorithm for computation on massively parallel accelerators

The design of efficient algorithms for large-scale gas dynamics computations with hybrid (heterogeneous) computing systems whose high performance relies on massively parallel accelerators is addressed. A high-order accurate finite volume algorithm with polynomial reconstruction on unstructured hybrid meshes is used to compute compressible gas flows in domains of complex geometry. The basic operations of the algorithm are implemented in detail for massively parallel accelerators, including AMD and NVIDIA graphics processing units (GPUs). Major optimization approaches and a computation transfer technique are covered. The underlying programming tool is the Open Computing Language (OpenCL) standard, which performs on accelerators of various architectures, both existing and emerging.