High Order Finite Volume Schemes Research Articles

A very fast high-order flux reconstruction for Finite Volume schemes for Computational Aeroacoustics

AbstractGiven the small wavelengths and wide range of frequencies of the acoustic waves involved in Aeroacoustics problems, the use of very accurate, low-dissipative numerical schemes is the only valid option to accurately capture these phenomena. However, as the order of the scheme increases, the computational time also increases. In this work, we propose a new high-order flux reconstruction in the framework of finite volume (FV) schemes for linear problems. In particular, it is applied to solve the Linearized Euler Equations, which are widely used in the field of Computational Aeroacoustics. This new reconstruction is very efficient and well suited in the context of very high-order FV schemes, where the computation of high-order flux integrals are needed at cell edges/faces. Different benchmark test cases are carried out to analyze the accuracy and the efficiency of the proposed flux reconstruction. The proposed methodology preserves the accuracy while the computational time relatively reduces drastically as the order increases.

Quinpi: Integrating Stiff Hyperbolic Systems with Implicit High Order Finite Volume Schemes

Quinpi: Integrating Stiff Hyperbolic Systems with Implicit High Order Finite Volume Schemes

A third-order compact finite volume scheme on unstructured grid for fluid flows

The large reconstruction stencil is one of the biggest obstacles in design and implementation of high-order finite volume schemes on unstructured grid. To tackle this problem, a third-order compact reconstruction scheme on unstructured grid is developed on the basis of compact least-squares (CLS) finite volume method. In building the reconstruction polynomial, the general boundary condition is adopted to ensure the unified third-order accuracy in both the internal and boundary cells. The reconstruction polynomial is solved in a semi-implicit sense to ensure high computational efficiency without solving the global linear algebraic equations. The theoretical accuracy order and stability limit are firstly verified by the Fourier analysis. The numerical accuracy and efficiency are then verified and validated with several incompressible and compressible flow cases. This work provides a reference to design and analysis of high-order compact finite volume schemes on unstructured grid with unified accuracy.

A general positivity-preserving algorithm for implicit high-order finite volume schemes solving the Euler and Navier-Stokes equations

This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes that solve compressible Euler and Navier-Stokes equations to ensure the positivity of density and internal energy (or pressure). Previous positivity-preserving algorithms are mainly based on the slope limiting or flux limiting technique, which rely on the existence of low-order positivity-preserving schemes. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes since it is difficult to know if a low-order implicit scheme is positivity-preserving. In the present paper, a new positivity-preserving algorithm is proposed in terms of the flux correction technique. And the factors of the flux correction are determined by a residual correction procedure. For a finite volume scheme that is capable of achieving a converged solution, we show that the correction factors are in the order of unity with additional high-order terms corresponding to the spatial and temporal rates of convergence. Therefore, the proposed positivity-preserving algorithm is accuracy-reserving and asymptotically consistent. The notable advantage of this method is that it does not rely on the existence of low-order positivity-preserving baseline schemes. Therefore, it can be applied to the implicit schemes solving Euler and especially Navier-Stokes equations. In the present paper, the proposed technique is applied to an implicit dual time-stepping finite volume scheme with temporal second-order and spatial high-order accuracy. The present positivity-preserving algorithm is implemented in an iterative manner to ensure that the dual time-stepping iteration will converge to the positivity-preserving solution. Another similar correction technique is also proposed to ensure that the solution remains positivity-preserving at each sub-iteration. Numerical results demonstrate that the proposed algorithm preserves positive density and internal energy in all test cases and significantly improves the robustness of the numerical schemes.

High-order genuinely multidimensional finite volume methods via kernel-based WENO

In this paper a family of fully multidimensional kernel-based reconstruction schemes for use in finite volume methods (FVMs) will be presented. These methods are intended for use in shock dominated problems, and stability is achieved through a suitable adaptation of the Adaptive Order Weighted Essentially Non-Oscillatory (WENO-AO) method to the proposed kernel-based reconstruction schemes. There are a number of key difficulties in the design of high-order finite volume schemes which will be discussed and addressed. High (4th and 6th) order convergence will be demonstrated on smooth exact solutions of the ideal MHD equations. The very same scheme will then be applied to extremely stringent astrophysical benchmark problems.

Об устойчивости и точности конечно-объёмных схем на неравномерных сетках

The paper studies the behavior of high-order finite-volume schemes for the 1D transport equation on non-uniform meshes. We consider finite-volume schemes with a polynomial reconstruction and schemes based on divided differences. We prove a sufficient stability condition on mildly deformed meshes and establish estimates for the solution error.

Improved higher-order finite volume scheme and its convergence analysis for collisional breakage equation

A new volume and number consistent finite volume scheme for the numerical solution of a collisional nonlinear breakage problem is introduced. The number consistency is achieved by introducing a single weight function in the flux formulation of finite volume scheme, whereas existing schemes for a linear fragmentation equation [Kumar et al. SIAM J. Numer. Anal. 53 (4), 1672-1689] and standard collisional nonlinear breakage equation [Das et al. SIAM J. Sci. Comp. 42 (6), B1570-B1598] require two weights for preserving both volume and number of particles. The higher efficiency and robustness of the proposed scheme allow it to be easily coupled with computational fluid dynamics (CFD) softwares such as COMSOL, Ansys and gPROMS, which is currently one of the predominant topics of discussion in particle technology. Consistency and stability via Lipschitz criterion are studied in detail to demonstrate second order convergence rate for the proposed scheme irrespective of both breakage kernel and nature of grids. Several benchmark problems are solved and validated against its analytical solution to analyze the accuracy of the new scheme.

Imposing slip conditions on curved boundaries for 3D incompressible flows with a very high-order accurate finite volume scheme on polygonal meshes

The conventional no-slip boundary condition does not always hold in several fluid flow applications and must be replaced with appropriate slip conditions according to the wall and fluid properties. However, not only slip boundary conditions are still a subject of discussion among fluid dynamicists, but also their numerical treatment is far from being well-developed, particularly in the context of very high-order accurate methods. The complexity of these conditions significantly increases when the boundary is not aligned with the chosen coordinate system and, even more challenging, when the fluid slips along a curved boundary. The present work proposes a simple and efficient numerical treatment of general slip boundary conditions on arbitrary curved boundaries for three-dimensional fluid flow problems governed by the incompressible Navier–Stokes equations. In that regard, two critical challenges arise: (i) achieving very high-order of convergence with arbitrary curved boundaries for the classical no-slip boundary conditions and (ii) extending the developed numerical techniques to impose general slip boundary conditions. The conventional treatment of curved boundaries relies on generating curved meshes to eliminate the geometrical mismatch between the physical and computational boundaries and achieve high-order of convergence. However, such an approach requires sophisticated meshing algorithms, cumbersome quadrature rules on curved elements, and complex non-linear transformations. In contrast, the reconstruction for off-site data (ROD) method handles arbitrary curved boundaries approximated with linear piecewise elements while employing polynomial reconstructions with specific linear constraints to fulfil the prescribed boundary conditions. For that purpose, the general slip boundary conditions are reformulated on a local orthonormal basis to allow a straightforward application of the ROD method with scalar boundary conditions. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed solely on the polygonal mesh elements. Several benchmark test cases of fluid flow problems in non-trivial three-dimensional curved domains are addressed and confirm that the proposed method effectively achieves up to the eighth-order of convergence.

The Compact and Accuracy Preserving Limiter for High-Order Finite Volume Schemes Solving Compressible Flows

The Compact and Accuracy Preserving Limiter for High-Order Finite Volume Schemes Solving Compressible Flows

A new family of semi-implicit Finite Volume/Virtual Element methods for incompressible flows on unstructured meshes

We introduce a new family of high order accurate semi-implicit schemes for the solution of nonlinear time-dependent systems of partial differential equations (PDE) on unstructured polygonal meshes. The time discretization is based on a splitting between explicit and implicit terms that may arise either from the multi-scale nature of the governing equations, which involve both slow and fast scales, or in the context of projection methods, where the numerical solution is projected onto the physically meaningful solution manifold. We propose to use a high order finite volume (FV) scheme for the explicit terms, hence ensuring conservation property and robustness across shock waves, while the virtual element method (VEM) is employed to deal with the discretization of the implicit terms, which typically requires an elliptic problem to be solved. The numerical solution is then transferred via suitable L2 projection operators from the FV to the VEM solution space and vice-versa. High order time accuracy is then achieved using the semi-implicit IMEX Runge–Kutta schemes, and the novel schemes are proven to be asymptotic preserving (AP) and well-balanced (WB). As representative models, we choose the shallow water equations (SWE), thus handling multiple time scales characterized by a different Froude number, and the incompressible Navier–Stokes equations (INS), which are solved at the aid of a projection method to satisfy the solenoidal constraint of the velocity field. Furthermore, an implicit discretization for the viscous terms is also devised for the INS model, which is based on the VEM technique. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the celerity nor on the viscous eigenvalues. A large suite of test cases demonstrates the accuracy and the capabilities of the new family of schemes to solve relevant benchmarks in the field of incompressible fluids.

Is the classic convex decomposition optimal for bound-preserving schemes in multiple dimensions?

Since proposed in Zhang and Shu (2010) [1], the Zhang–Shu framework has attracted extensive attention and motivated many bound-preserving (BP) high-order discontinuous Galerkin and finite volume schemes for various hyperbolic equations. A key ingredient in the framework is the decomposition of the cell averages of the numerical solution into a convex combination of the solution values at certain quadrature points, which helps to rewrite high-order schemes as convex combinations of formally first-order schemes. The classic convex decomposition originally proposed by Zhang and Shu has been widely used over the past decade. It was verified, only for the 1D quadratic and cubic polynomial spaces, that the classic decomposition is optimal in the sense of achieving the mildest BP CFL condition. Yet, it remained unclear whether the classic decomposition is optimal in multiple dimensions. In this paper, we find that the classic multidimensional decomposition based on the tensor product of Gauss–Lobatto and Gauss quadratures is generally not optimal, and we discover a novel alternative decomposition for the 2D and 3D polynomial spaces of total degree up to 2 and 3, respectively, on Cartesian meshes. Our new decomposition allows a larger BP time step-size than the classic one, and moreover, it is rigorously proved to be optimal to attain the mildest BP CFL condition, yet requires much fewer nodes. The discovery of such an optimal convex decomposition is highly nontrivial yet meaningful, as it may lead to an improvement of high-order BP schemes for a large class of hyperbolic or convection-dominated equations, at the cost of only a slight and local modification to the implementation code. Several numerical examples are provided to further validate the advantages of using our optimal decomposition over the classic one in terms of efficiency.

A MOOD-like compact high order finite volume scheme with adaptive mesh refinement

In this paper, a novel Finite Volume (FV) scheme for obtaining high order approximations of solutions of multi-dimensional hyperbolic systems of conservation laws within an Adaptive Mesh Refinement framework is proposed. It is based on a point-wise polynomial reconstruction that avoids the recalculation of reconstruction stencils and matrices whenever a mesh is refined or coarsened. It also couples both the limiting of the FV scheme and the refinement procedure, taking advantage of the Multi-dimensional Optimal Order Detection (MOOD) detection criteria. The resulting computational procedure is employed to simulate test cases of increasing difficulty using two models of Partial Differential Equations: the Euler system and the radiative M1 model, thus demonstrating its efficiency.

A high order positivity-preserving conservative WENO remapping method based on a moving mesh solver

In this paper, a high order accurate positivity-preserving conservative remapping algorithm is developed. Quadrilateral meshes in two dimensions are used as examples. This remapping method is based on the numerical solution of the trivial equation ∂u∂t=0 on a moving mesh, which is the old mesh before remapping at t=0 and is the new mesh after remapping at t=T. A high order finite volume scheme on the moving mesh is used to solve this problem. Specifically, we adopt the multi-resolution weighted essentially non-oscillatory (WENO) method for the spatial discretization and a strong stability preserving (SSP) Runge-Kutta method for the temporal discretization. The remapping algorithm is high order accurate under very mild smoothness requirement (Lipschitz continuity) on the mesh movement velocity, which can always be satisfied with a suitable choice of the final pseudo-time T. Furthermore, we design our remapping algorithm to have positivity-preserving property by using the linear scaling positivity-preserving limiter so that the algorithm could ensure the positivity-preserving property of relevant physical variables and maintain conservation and original order of accuracy. A series of numerical experiments are given to demonstrate the properties of our remapping algorithm such as high order accuracy, essentially non-oscillatory performance, positivity-preserving and high computational efficiency.

A Very Easy High-Order Well-Balanced Reconstruction for Hyperbolic Systems with Source Terms

When adopting high-order finite volume schemes based on MUSCL reconstruction techniques to approximate the weak solutions of hyperbolic systems with source terms, the preservation of the steady states turns out to be very challenging. Indeed, the designed reconstruction must preserve the steady states under consideration in order to get the required well-balancedness property. A priori, to capture such a steady state, one needs to solve some strongly nonlinear equations. Here, we design a very easy correction to high-order finite volume methods. This correction can be applied to any scheme of order greater than or equal to 2, such as a MUSCL-type scheme, and ensures that this scheme exactly preserves the steady solutions. In contrast to usual techniques, our scheme avoids the inversion of the nonlinear function that governs the steady solutions. Moreover, for underdetermined steady solutions, several nonlinear functions must be considered simultaneously. Since the derived correction only considers the evaluation of the governing nonlinear functions, we are able to deal with underdetermined stationary systems. Several numerical experiments illustrate the relevance of the proposed well-balanced correction, as well as its main limitation, namely the fact that it may fail at being both well-balanced and more than second-order accurate for a specific class of initial conditions.

A new mixed Boltzmann-BGK model for mixtures solved with an IMEX finite volume scheme on unstructured meshes

In this work, we consider a novel model for a binary mixture of inert gases. The model, which preserves the structure of the original Boltzmann equations, combines integro-differential collision operators with BGK relaxation terms in each kinetic equation: the first involving only collisions among particles of the same species, while the second ones taking into account the inter–species interactions. We prove consistency of the model: conservation properties, positivity of all temperatures, H-theorem and convergence to a global equilibrium in the shape of a global Maxwell distribution. We also derive hydrodynamic equations under different collisional regimes. In a second part, to numerically solve the governing equations, we introduce a class of time and space high order finite volume schemes that are able to capture the behaviors of the different hydrodynamic limit models: the classical Euler equations as well as the multi-velocities and temperatures Euler system. The methods work by integrating the distribution functions over arbitrarily shaped and closed control volumes in 2D using Central Weighted ENO (CWENO) techniques and make use of spectral methods for the approximation of the Boltzmann integrals with high order Implicit-Explicit (IMEX) Runge Kutta schemes. For these methods, we prove accuracy and preservation of the discrete asymptotic states. In the numerical section we first show that the methods indeed possess the theoretical order of accuracy for different regimes and second we analyse their capacity in solving different two dimensional problems arising in kinetic theory. To speed up the computational time, all simulations are run with MPI parallelization on 64 cores, thus showing the potentiality of the proposed methods to be used for HPC (High Performance Computing) on massively parallel architectures.

High-order compact finite volume schemes for solving the Reynolds averaged Navier-Stokes equations on the unstructured mixed grids with a large aspect ratio

In this paper, high-order compact finite volume schemes on the unstructured grids based on the variational reconstruction are developed to solve the Reynolds averaged Navier-Stokes equations closed by the Spalart-Allmaras one-equation turbulence model. Encouraging progress is made in addressing the following two challenging problems: reducing the numerical errors on the large aspect ratio grids and avoiding the negative turbulent viscosity associated with the high-order methods. On grids with large aspect ratios, a three-step procedure is designed to optimize the functional parameters of variational reconstruction. In addition, an exponential decay procedure is proposed to cure the negative turbulent viscosity problem of the Spalart-Allmaras model. The exponential decay procedure has the advantage of being able to be used with any spatial discretization method and with the implicit temporal discretization. Numerical tests show significant benefits of the high-order schemes in predicting the skin frictions, capturing some important flow structures, and achieving grid-independent solutions. The numerical tests also show that the proposed schemes are sufficiently robust for practical applications.

A new class of higher-ordered/extended Boussinesq system for efficient numerical simulations by splitting operators

In this work, we numerically study the higher-ordered/extended Boussinesq system describing the propagation of water-waves over flat topography. A reformulation of the same order of precision that avoids the calculation of high order derivatives on the surface deformation is proposed. We show that this formulation enjoys an extended range of applicability while remaining stable. Moreover, a significant improvement in terms of linear dispersive properties in high frequency regime is made due to the suitable adjustment of a dispersion correction parameter. 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 under two main goals: validating the model and the numerical methods and assessing the potential need of such higher-order model. The applicability of the proposed model and numerical method in practical problems is illustrated by a comparison with experimental data.

A bound-preserving and positivity-preserving finite volume WENO scheme for solving five-equation model of two-medium flows

In this paper, we develop a physical-constraint-preserving high-order finite volume WENO scheme for compressible two-medium gas–gas and gas–liquid interfacial flows. Using high-order WENO reconstruction and Lax–Friedrichs approximate Riemann flux, a high-order finite volume scheme for solving the five-equation transport model is presented by carefully designing the discretization of the non-conservative term. A novel and thorough analysis for deriving the sufficient condition to obtain physically admissible solutions, including bounded volume fractions, positive partial densities, internal energy and square of sound speed, is proposed and studied in details. Then, an effective bound- and positivity-preserving limiting procedure is developed based on the finite volume framework. Moreover, the proposed bound- and positivity-preserving limiting procedure is also applicable to a special type of hyperbolic tangent function as local solution profile, which is adopted to control numerical dissipation and sharpen material interfaces in this work. Typical problems of compressible gas–gas and gas–liquid two-medium flows are investigated to demonstrate the performance and capability of the proposed scheme. Numerical results demonstrate that the bound- and positivity-preserving limiting strategy is of vital importance in enhancing the robustness for high-order methods under severe flow conditions.

Very high-order accurate finite volume scheme for the steady-state incompressible Navier–Stokes equations with polygonal meshes on arbitrary curved boundaries

The numerical solution of the incompressible Navier–Stokes equations raises challenging numerical issues on the development of accurate and robust discretisation techniques due to the div-grad duality. Significant advances have been achieved in the context of high-order of convergence methods, but many questions remain unsolved in the quest of practical, accurate, and efficient approaches. In particular, the discretisation of problems with arbitrary curved boundaries has been the subject of intensive research, where the conventional practice consists in employing curved meshes to avoid accuracy deterioration. The present work proposes a novel approach, the reconstruction for off-site data method, which employs polygonal meshes to approximate arbitrary smooth curved boundaries with linear piecewise elements. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed on the polygonal mesh elements. Boundary conditions are taken into account via polynomial reconstructions with specific linear constraints defined for a set of points on the physical boundary. The very-high order of convergence is preserved without relying on the full curve parametrisation, avoiding the limitations of curved mesh approaches, such as sophisticated mesh generation algorithms, cumbersome quadrature rules, and complex non-linear transformations. A comprehensive verification benchmark is provided, with numerical test cases for several fluid flow problems, to demonstrate the capability of the proposed approach to achieve very high-orders of convergence.

Compact multi-stage reconstruction method on polyhedral unstructured grids: Extension to higher-order finite volume scheme

In this paper, a novel high-order finite volume scheme is proposed for arbitrary unstructured grids based on multi-stage reconstruction procedure using a compact stencil. Unlike the conventional approaches such as k-exact formulation, the coefficients of the reconstruction polynomial are determined by using a deferred correction algorithm which is built on the relations that make use of not only the volume integrated averages but also the spatial derivatives on the neighboring cells. The newly designated FVMS4 (Finite Volume method based on Multi-Stage procedure with 4th-order reconstruction) substantially improves the solution accuracy in contrast to its previous variant, FVMS3, without extending the reconstruction stencil, which allows FVMS4 to be formulated with algorithmic simplicity and numerical efficiency. In order to eliminate the numerical oscillations associated with high-order schemes, it is combined with a limiting projection scheme AOD (Adaptive Order Detection) to choose the optimal degree of reconstruction polynomial for trouble cells that may contain the discontinuous solutions. The resulting reconstruction method, namely FVMS4-AOD scheme achieves essentially non-oscillatory, less-dissipative and highly accurate results for smooth and discontinuous solutions. Its performance has been verified by solving linear and nonlinear hyperbolic systems for various benchmark tests in 2 and 3 dimensions. The numerical results substantiate that the proposed method provides an innovative numerical formulation capable of handling both complex physics and geometries which has significant benefits for practical applications.