High Order Finite Volume Schemes Research Articles

On the dissipation mechanism of lattice Boltzmann method when modeling 1-d and 2-d water hammer flows

This paper uses the multiple relaxation times lattice Boltzmann method (LBM) to model acoustic wave propagation in water-filled conduits (i.e. water hammer applications). The LBM scheme is validated by solving some classical water hammer (WH) numerical tests in one-dimensional and two-dimensional cases. In addition, this work discusses the accuracy, stability and robustness of LBM when solving high frequency (i.e. radial modes are excited) acoustic waves in water-filled pipes. The results are compared with a high order finite volume scheme based on Riemann Solver. The results show that LBM is capable to model WH applications with high accuracy and performance at low frequency cases; however, it loses stability and performance for high frequency (HF) cases. It is discussed that this is due to the neglected high order terms of the equivalent macroscopic equations considered for LB numerical formulation.

High-order compact finite volume methods on unstructured grids with adaptive mesh refinement for solving inviscid and viscous flows

In the present paper, high-order finite volume schemes on unstructured grids developed in our previous papers are extended to solve three-dimensional inviscid and viscous flows. The high-order variational reconstruction technique in terms of compact stencil is improved to reduce local condition numbers. To further improve the efficiency of computation, the adaptive mesh refinement technique is implemented in the framework of high-order finite volume methods. Mesh refinement and coarsening criteria are chosen to be the indicators for certain flow structures. One important challenge of the adaptive mesh refinement technique on unstructured grids is the dynamic load balancing in parallel computation. To solve this problem, the open-source library p4est based on the forest of octrees is adopted. Several two- and three-dimensional test cases are computed to verify the accuracy and robustness of the proposed numerical schemes.

ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics

We present a new class of high-order accurate numerical algorithms for solving the equations of general-relativistic ideal magnetohydrodynamics in curved spacetimes. In this paper we assume the background spacetime to be given and static, i.e., we make use of the Cowling approximation. The governing partial differential equations are solved via a new family of fully-discrete and arbitrary high-order accurate path-conservative discontinuous Galerkin (DG) finite-element methods combined with adaptive mesh refinement and time accurate local timestepping. In order to deal with shock waves and other discontinuities, the highorder DG schemes are supplemented with a novel a-posteriori subcell finite-volume limiter, which makes the new algorithms as robust as classical second-order total-variation diminishing finite-volume methods at shocks and discontinuities, but also as accurate as unlimited high-order DG schemes in smooth regions of the flow. We show the advantages of this new approach by means of various classical two- and three-dimensional benchmark problems on fixed spacetimes. Finally, we present a performance and accuracy comparisons between Runge-Kutta DG schemes and ADER high-order finite-volume schemes, showing the higher efficiency of DG schemes.

Extended bounds limiter for high-order finite-volume schemes on unstructured meshes

This paper explores the impact of the definition of the bounds of the limiter proposed by Michalak and Ollivier-Gooch in [56] (2009), for higher-order Monotone-Upstream Central Scheme for Conservation Laws (MUSCL) numerical schemes on unstructured meshes in the finite-volume (FV) framework. A new modification of the limiter is proposed where the bounds are redefined by utilising all the spatial information provided by all the elements in the reconstruction stencil. Numerical results obtained on smooth and discontinuous test problems of the Euler equations on unstructured meshes, highlight that the newly proposed extended bounds limiter exhibits superior performance in terms of accuracy and mesh sensitivity compared to the cell-based or vertex-based bounds implementations.

Bound-preserving modified exponential Runge–Kutta discontinuous Galerkin methods for scalar hyperbolic equations with stiff source terms

In this paper, we develop bound-preserving modified exponential Runge–Kutta (RK) discontinuous Galerkin (DG) schemes to solve scalar hyperbolic equations with stiff source terms by extending the idea in Zhang and Shu [43]. Exponential strong stability preserving (SSP) high order time discretizations are constructed and then modified to overcome the stiffness and preserve the bound of the numerical solutions. It is also straightforward to extend the method to two dimensions on rectangular and triangular meshes. Even though we only discuss the bound-preserving limiter for DG schemes, it can also be applied to high order finite volume schemes, such as weighted essentially non-oscillatory (WENO) finite volume schemes as well.

Very high-order accurate finite volume scheme on curved boundaries for the two-dimensional steady-state convection–diffusion equation with Dirichlet condition

• Very high-order finite volume scheme for curved geometries. • Reconstruction for Off-site Data to provide the optimal order with curved domain. • No curved element and complex mapping on reference element. • A set numerical experiences with non-trivial geometries to assess the method order. Accuracy may be dramatically reduced when the boundary domain is curved and numerical schemes require a specific treatment of the boundary condition to preserve the optimal order. In the finite volume context, Ollivier-Gooch and Van Altena (2002) has proposed a technique to overcome such limitation and restore the very high-order accuracy which consists in specific restrictions considered in the least-squares minimization associated to the polynomial reconstruction. The method suffers from several drawbacks, particularly, the use of curved elements that requires sophisticated meshing algorithms. We propose a new method where the physical domain and the computational domain are distinct and we introduce the Reconstruction for Off-site Data (ROD) where polynomial reconstructions are carried out on the mesh using data localized outside of the computational domain, namely the Dirichlet condition situated on the physical domain. A series of numerical tests assess the accuracy, convergence rates, robustness, and efficiency of the new method and show that the boundary condition is fully integrated in the scheme with a very high-order accuracy and the optimal convergence rate is achieved.

High order finite volume schemes for balance laws with stiff relaxation

The paper deals with the construction and analysis of efficient high order finite volume shock capturing schemes for the numerical solution of hyperbolic systems with stiff relaxation. In standard high order finite volume schemes it is difficult to treat the average of the source implicitly, since the computation of such average couples neighboring cells, making implicit schemes extremely expensive. The main novelty of the paper is that the average of the source is split into the sum of the source evaluated at the cell average plus a correction term. The first term is treated implicitly, while the small correction is treated explicitly, using IMEX-Runge–Kutta methods, thus resulting in a very effective semi-implicit scheme. This approach allows the construction of effective high order schemes in space and time. An asymptotic analysis is performed for small values of the relaxation parameter, giving an indication on the structure of the IMEX schemes that have to be adopted for time discretization. Several numerical tests confirm the accuracy and efficiency of the approach.

A Multi-State HLL Approximate Riemann Solver for Solid/Vacuum Riemann Problem

A new multi-state HLLD (‘‘D’’ stands for Discontinuities.) approximate Riemann solver for Riemann problem of nonlinear elastic solid is developed based on the assumption that a wave configuration for the solution that consists of five waves (two slow waves, two fast waves and a contact discontinuity) separating six constant states. Since the intermediate states satisfied with the Rankine-Hugoniot relations in this approximate Riemann system are analytically obtained, the HLLD Riemann solver can be constructed straightforwardly. The Piecewise Parabolic Method (PPM) is used directly to construct high-order finite-volume schemes. Numerical tests demonstrate that the scheme PPM coupled with HLLD is robust and efficient. It indicates that the scheme PPM+ HLLD can be useful in practical applications for the non-linear elasticity.

High-frequency acoustic wave properties in a water-filled pipe. Part 1: dispersion and multi-path behaviour

ABSTRACTDefect resolution using transient-based detection methods (TBDDM) improves as frequency bandwidth of injected signals increases. High-frequency waves (HFW) excite radial and azimuthal modes, however, making the 1D water-hammer ineffective for these phenomena. This paper focuses on idealized HFW in inviscid, slightly compressible fluid in rigid, unbounded, circular conduits. HFW waves are generated by an axisymmetric wave source. High-order 2D finite volume scheme is developed for HFW. Results show that injected probing waves containing a cut-off frequency exhibit significant wave dispersion, and group velocity varies over a wide range of speeds, resulting in wave-energy spatial spreading with large amplitude reduction. Injected signals without any cut-off frequency remain spatially separated. Size of wave source significantly affects energy distribution within wave modes. Practical implications are that probing wave and defect resolution effectiveness may be increased using (i) probing HFW without cut-off frequencies contained in the bandwidth of the injected probing signal, and (ii) larger power sources for the injection of probing wave signals.

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.

High-order finite-volume solutions of the steady-state advection–diffusion equation with nonlinear Robin boundary conditions

We propose high-order finite-volume schemes for numerically solving the steady-state advection–diffusion equation with nonlinear Robin boundary conditions. Although the original motivation comes from a mathematical model of blood clotting, the nonlinear boundary conditions may also apply to other scientific problems. The main contribution of this work is a generic algorithm for generating third-order, fourth-order, and even higher-order explicit ghost-filling formulas to enforce nonlinear Robin boundary conditions in multiple dimensions. Under the framework of finite volume methods, this appears to be the first algorithm of its kind. Numerical experiments on boundary value problems show that the proposed fourth-order formula can be much more accurate and efficient than a simple second-order formula. Furthermore, the proposed ghost-filling formulas may also be useful for solving other partial differential equations.

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.

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.

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.

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.

Cell centered direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity

This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics proposed by Peshkov & Romenski [Peshkov I, Romenski E. A hyperbolic model for viscous Newtonian flows. Continuum Mechanics and Thermodynamics 2016;28:85–104.], which is based on the theory of nonlinear hyperelasticity of Godunov & Romenski [67][Godunov S, Romenski E. Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates. Journal of Applied Mechanics and Technical Physics 1972;13:868–885.][Godunov S., Romenski E., Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/ Plenum Publishers; 2003.], further denoted by GPR model. Notably, the governing PDE system is symmetric hyperbolic and fully consistent with the first and the second principle of thermodynamics. The nonlinear system of governing equations of the GPR model is overdetermined, large and includes stiff source terms as well as non-conservative products. In this paper we solve this model for the first time on moving unstructured meshes in multiple space dimensions by employing high order accurate one-step ADER-WENO finite volume schemes in the context of cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) algorithms.The numerical method is based on a WENO polynomial reconstruction operator on moving unstructured meshes, a fully-discrete one-step ADER scheme that is able to deal with stiff sources [Dumbser M., Enaux C., Toro E., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. Journal of Computational Physics . 2008a;227:3971–4001.], a nodal solver with relaxation to determine the mesh motion, and a path-conservative technique of Castro & Parés for the treatment of non-conservative products [Parés C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM Journal on Numerical Analysis. 2006;44:300–321.][Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems. Mathematics of Computation 2006;75:1103–1134.]. We present numerical results obtained by solving the GPR model with ADER-WENO-ALE schemes in the stiff relaxation limit, showing that fluids (Euler or Navier-Stokes limit), as well as purely elastic or elasto-plastic solids can be simulated in the framework of nonlinear hyperelasticity with the same system of governing PDE. The obtained results are in good agreement when compared to exact or numerical reference solutions available in the literature.

High-order central Hermite WENO schemes: Dimension-by-dimension moment-based reconstructions

In this paper, a class of high-order central finite volume schemes is proposed for solving one- and two-dimensional hyperbolic conservation laws. Formulated on staggered meshes, the methods involve Hermite WENO (HWENO) spatial reconstructions, and Lax–Wendroff type discretizations or the natural continuous extension of Runge–Kutta methods in time. Differently from the central Hermite WENO methods we developed previously in Tao et al. (2015) [34] , the spatial reconstructions, a core ingredient of the methods, are based on the zeroth-order and the first-order moments of the solution, and are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. This leads to much simpler implementation of the methods in higher dimension and better cost efficiency. Meanwhile, the proposed methods have the attractive features of the general central Hermite WENO methods such as being compact in reconstruction and requiring neither flux splitting nor numerical fluxes, while being accurate and essentially non-oscillatory. A collection of one- and two-dimensional numerical examples is presented to demonstrate high resolution and robustness of the methods in capturing smooth and non-smooth solutions.

Bound-preserving discontinuous Galerkin methods for relativistic hydrodynamics

In this paper, we develop a discontinuous Galerkin (DG) method to solve the ideal special relativistic hydrodynamics (RHD) and design a bound-preserving (BP) limiter for this scheme by extending the idea in X. Zhang and C.-W. Shu, (2010) [56]. For RHD, the density and pressure are positive and the velocity is bounded by the speed of light. One difficulty in numerically solving the RHD in its conservative form is that the failure of preserving these physical bounds will result in ill-posedness of the problem and blowup of the code, especially in extreme relativistic cases. The standard way in dealing with this difficulty is to add extra numerical dissipation, while in doing so there is no guarantee of maintaining the high order of accuracy. Our BP limiter has the following features. It can theoretically guarantee to preserve the physical bounds for the numerical solution and maintain its designed high order accuracy. The limiter is local to the cell and hence is very easy to implement. Moreover, it renders L1-stability to the numerical scheme. Numerical experiments are performed to demonstrate the good performance of this bound-preserving DG scheme. Even though we only discuss the BP limiter for DG schemes, it can be applied to high order finite volume schemes, such as weighted essentially non-oscillatory (WENO) finite volume schemes as well.

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.

A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation

Abstract A new very high-order finite volume method to solve problems with harmonic and biharmonic operators for onedimensional geometries is proposed. The main ingredient is polynomial reconstruction based on local interpolations of mean values providing accurate approximations of the solution up to the sixth-order accuracy. First developed with the harmonic operator, an extension for the biharmonic operator is obtained, which allows designing a very high-order finite volume scheme where the solution is obtained by solving a matrix-free problem. An application in elasticity coupling the two operators is presented. We consider a beam subject to a combination of tensile and bending loads, where the main goal is the stress critical point determination for an intramedullary nail.