Finite Volume Weighted Essentially Non-oscillatory Schemes Research Articles

Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-conservative Products

Higher order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes have been constructed for conservation laws. For multidimensional problems, they offer high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of comparable accuracy. This makes them quite attractive for several science and engineering applications. But, to the best of our knowledge, such schemes have not been extended to non-linear hyperbolic systems with non-conservative products. In this paper, we perform such an extension which improves the domain of applicability of such schemes. The extension is carried out by writing the scheme in fluctuation form. We use the HLLI Riemann solver of Dumbser and Balsara (2016) as a building block for carrying out this extension. Because of the use of an HLL building block, the resulting scheme has a proper supersonic limit. The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme, thus expanding its domain of applicability. Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions, making it very easy for users to transition over to the present formulation. For conservation laws, the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO, with two major advantages:- 1) It can capture jumps in stationary linearly degenerate wave families exactly. 2) It only requires the reconstruction to be applied once. Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of accuracy for smooth multidimensional flows. Stringent Riemann ... *Abstract truncated, see PDF*

Constrained least-squares based adaptive-order finite-volume WENO scheme for the simulation of viscous compressible flows on unstructured grids

We develop a novel constrained least-squares technique for fourth-order weighted essentially non-oscillatory (WENO) polynomial reconstruction on general unstructured grids composed of curved-edged quadrilateral elements. Our technique relies on a segregation of the conditions that match the spatial averages of the candidate WENO polynomial and the reconstructed state variable over stencil-forming cells into two sets. The first set consists of exactly enforced linearly independent constraints that are applied only over the central cell and its edge-sharing neighbors. The second set consists of the remaining conditions and are enforced only approximately in a least-squares sense. We develop a specialized technique that reduces the computational burden associated with the numerical solution of the constrained least-squares system for the WENO interpolation coefficients. For a comparable computational expense, our constrained reconstruction approach yields pointwise estimates that are consistently more accurate than the conventional least-squares based estimates over a wide spectrum of spatial wavenumbers. This enhanced spectral resolution yields a significant, over two-fold increase in the accuracy with which unsteady smooth flow features, such as an advecting isentropic vortex, are captured over highly distorted curvilinear unstructured grids. Our constraint based reconstruction framework is generic, facilitates implementation of high-order boundary closures, and is amenable to large scale parallelization over several thousand CPU cores. Numerical tests over a range of inviscid and viscous flow configurations are presented to demonstrate the accuracy and robustness of our technique. Our constrained least-squares based adaptive WENO discretization faithfully captures intricate features that arise in strongly shocked, and highly unsteady and separated complex geometry flows. Our results suggest exact preservation of the match between the cell averages of the interpolant and the reconstructed state variable over the nearest neighbors to be quite an effective strategy that dramatically enhances the accuracy and spectral resolution of the high-order finite-volume reconstruction operation.

Positivity-preserving high order finite difference WENO schemes for compressible Navier-Stokes equations

In this paper, we construct a high order weighted essentially non-oscillatory (WENO) finite difference discretization for compressible Navier-Stokes (NS) equations, which is rendered positivity-preserving of density and internal energy by a positivity-preserving flux splitting and a scaling positivity-preserving limiter. The novelty of this paper is WENO reconstruction performed on variables from a positivity-preserving convection diffusion flux splitting, which is different from conventional WENO schemes solving compressible NS equations. The core advantages of our proposed method are robustness and efficiency, which especially are suitable for solving tough demanding problems of both compressible Euler and NS equation including low density and low pressure flow regime. Moreover, in terms of computational cost, it is more efficient and easier to implement and extend to multi-dimensional problems than the positivity-preserving high order discontinuous Galerkin schemes and finite volume WENO scheme for solving compressible NS equations on rectangle domain. Benchmark tests demonstrate that the proposed positivity-preserving WENO schemes are high order accuracy, efficient and robust without excessive artificial viscosity for demanding problems involving with low density, low pressure, and fine structure.

A fourth-order conservative semi-Lagrangian finite volume WENO scheme without operator splitting for kinetic and fluid simulations

In this paper, we present a fourth-order conservative semi-Lagrangian (SL) finite volume (FV) weighted essentially non-oscillatory (WENO) scheme without operator splitting for two-dimensional linear transport equations with applications in kinetic models including the nonlinear Vlasov–Poisson system, the guiding center Vlasov model and the incompressible Euler equation in the vorticity-stream function formulation. To achieve fourth-order accuracy in space, two main ingredients are proposed in the SL FV formulation. Firstly, we introduce a so-called cubic-curved quadrilateral upstream cell and applying an efficient clipping method to evaluate integrals on upstream cells. Secondly, we construct a new WENO reconstruction operator, which recovers a P3 polynomial from neighboring cell averages. Mass conservation is accomplished with the mass conservative nature of the reconstruction operator and the SL formulation. A positivity-preserving limiter is applied to maintain the positivity of the numerical solution wherever appropriate. For nonlinear kinetic models, the SL scheme is coupled with a fourth-order Runge–Kutta exponential integrator for high-order temporal accuracy. Extensive benchmarks are tested to verify the designed properties.

A finite volume WENO scheme for immiscible inviscid two-phase flows

This article presents a two-phase finite volume (FV) method based on the concept of equilibrium volume fraction for simulating inviscid two-phase flows. A reduced three-equation hyperbolic system is adopted as the governing equations, which is closed with a barotropic equation of state for both phases. A modified weighted essentially non-oscillatory (WENO) scheme is proposed for reconstructing the fluid state at cell-interfaces, which can avoid strictly pressure or velocity oscillations near material interfaces in the test-case of the advection of an isolated interface, while retaining the convergence order of the WENO scheme in 1D and 2D cases. Besides, with the generalized Riemann invariants (GRI), the eigenstructure of the hyperbolic system is analyzed and an approximate Riemann flux solver is proposed based on the linearization of the GRI. The semi-discrete system is explicitly integrated in time by means of the classical 4th-order Runge-Kutta (RK4) scheme. The proposed scheme is validated in a series of two-phase test-cases, such as advection of a two-phase vortex, linear sloshing, long-time wave propagation and dam-break flows, for which good agreements are obtained with the references.

High order well-balanced finite difference WENO interpolation-based schemes for shallow water equations

A numerical framework of the generalized form of high order well-balanced finite difference weighted essentially non-oscillatory (WENO) interpolation-based schemes is proposed for the shallow water equations. It demonstrates more flexible construction process than the classical WENO reconstruction-based schemes. The weighted compact nonlinear schemes and finite difference alternative WENO schemes are two specific cases. To maintain the exact C-property, the splitting technique for the source term in the finite difference scheme [Xing and Shu, J. Comput. Phys. 208 (2005)] and the reconstruction technique in the finite volume WENO scheme [Xing and Shu, J. Comput. Phys. 214 (2006)] are adopted. The proposed scheme can be proved mathematically to maintain the exact C-property and demonstrates numerically that it is well-balanced by construction for the stationary water surface. Moreover, the local characteristic projections are employed to further mitigate the Gibbs oscillations. The proposed generic high order WENO schemes not only achieve high order accuracy but also capture the high gradients/shock waves essentially non-oscillatory. Meanwhile, the small perturbation problems can be resolved well on a coarse grid.

A new type of third-order finite volume multi-resolution WENO schemes on tetrahedral meshes

In this continuing paper of Zhu and Shu (2018) [62], Zhu and Shu (2019) [63], we design a new third-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws on tetrahedral meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation. Comparing with classical third-order finite volume WENO schemes Zhang and Shu (2009) [56] on tetrahedral meshes, the crucial advantages of such new multi-resolution WENO schemes are their simplicity and compactness with the application of only three unequal-sized central stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, their easy choice of arbitrary positive linear weights without considering the topology of the tetrahedral meshes, their optimal order of accuracy in smooth regions, and their suppression of spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO scheme can be any positive numbers on the condition that their sum is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite volume WENO scheme on tetrahedral meshes. By performing such new spatial reconstruction procedures and adopting a third-order TVD Runge-Kutta method for time discretization, the occupied memory is decreased and the computing efficiency is increased. This new third-order finite volume multi-resolution WENO scheme is suitable for large scale engineering applications and could maintain good convergence property for steady-state problems on tetrahedral meshes. Benchmark examples are computed to demonstrate the robustness and good performance of these new finite volume WENO schemes.

High-order gas-kinetic scheme with three-dimensional WENO reconstruction for the Euler and Navier-Stokes solutions

In this paper, a simple and efficient third-order weighted essentially non-oscillatory (WENO) reconstruction is developed for three-dimensional flows, in which the idea of two-dimensional WENO-AO scheme on unstructured meshes [42] is adopted. In the classical finite volume type WENO schemes, the linear weights for candidate stencils are obtained by solving linear systems at Gaussian quadrature points of cell interface. For the three-dimensional scheme, such operations at twenty-four Gaussian quadrature points of a hexahedron would reduce the efficiency greatly, especially for the moving-mesh computation. Another drawback of the classical WENO schemes is the appearance of negative weights with irregular local topology, which affect the robustness of spatial reconstruction. For the three-dimensional WENO-AO scheme, a simple strategy of selecting big stencil and sub-stencils for reconstruction is proposed. With the reconstructed quadratic polynomial from big stencil and linear polynomials from sub-stencils, the linear weights are chosen as positive numbers with the requirement that their summation equals to one and be independent of local mesh topology. With such WENO reconstruction, a high-order gas-kinetic scheme (HGKS) is developed for both three-dimensional inviscid and viscous flows. Taken the grid velocity into account, the scheme is extended into the moving-mesh computation as well. Numerical results are provided to illustrate the good performance of such new finite volume WENO scheme. In the future, such WENO reconstruction will be extended to the unstructured meshes.

Numerical issues in gas flow dynamics with hydraulic shocks using high order finite volume WENO schemes

Accurate and efficient simulation of the hydraulic shock phenomenon in pipeline systems is of paramount importance. Even though the conservation-law formulation of the governing equations is here strongly advocated, the nonconservative form is still frequently used. This also concerns its mathematical conservative form. We investigated the numerical consequences of using the compressible gas flow model in the latter form while simulating a hydraulic shock. In this context, we also solved two Riemann problems. For the investigation, we used the third-, fifth- and seventh-order accurate weighted essentially non-oscillatory (WENO) scheme along with the Lax–Friedrichs solver at the cell interfaces. Both the classical finite volume WENO scheme and its modification WENO–Z have been implemented. A procedure based on the method of manufactured solutions has been developed to verify whether the numerical code solved correctly the hyperbolic set of equations. We demonstrated that the solutions of the conservative and nonconservative formulations are similar if we have smooth variations in the solution domain. The convective inertia term in the momentum equation should not be ignored. In the presence of shocks, differences in oscillating behavior and slope steepness near the discontinuities were observed. For the hydraulic shock problem, spurious oscillations appeared while using the nonconservative formulation in combination with the WENO–Z reconstruction.

Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

A new type of high-order multi-resolution weighted essentially non-oscillatory (WENO) schemes (Zhu and Shu in J Comput Phys, 375: 659–683, 2018) is applied to solve for steady-state problems on structured meshes. Since the classical WENO schemes (Jiang and Shu in J Comput Phys, 126: 202–228, 1996) might suffer from slight post-shock oscillations (which are responsible for the residue to hang at a truncation error level), this new type of high-order finite-difference and finite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state simulations. This new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes, could obtain fifth-order, seventh-order, and ninth-order in smooth regions, and could gradually degrade to first-order so as to suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO schemes can be any positive numbers on the condition that their sum is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite-difference and finite-volume WENO schemes for solving steady-state problems. In comparison with the classical fifth-order finite-difference and finite-volume WENO schemes, the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems.

A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes

In this paper, we continue our work in [46] and propose a new type of high-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. Although termed “multi-resolution WENO schemes”, we only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. We construct new third-order, fourth-order, and fifth-order WENO schemes using three or four unequal-sized central spatial stencils, different from the classical WENO procedure using equal-sized biased/central spatial stencils for the spatial reconstruction. The new WENO schemes could obtain the optimal order of accuracy in smooth regions, and could degrade gradually to first-order of accuracy so as to suppress spurious oscillations near strong discontinuities. This is the first time that only a series of unequal-sized hierarchical central spatial stencils are used in designing arbitrary high-order finite volume WENO schemes on triangular meshes. The main advantages of these schemes are their compactness, robustness, and their ability to maintain good convergence property for steady-state computation. The linear weights of such WENO schemes can be any positive numbers on the condition that they sum to one. Extensive numerical results are provided to illustrate the good performance of these new finite volume WENO schemes.

Adaptive mapping for high order WENO methods

In this paper, a novel mapping approach through the use of adaptive mapping functions is introduced for high order weighted essentially non-oscillatory (WENO) methods. The new class of adaptive mapping functions are designed to adjust themselves to the solution based on a simple parameter calculated using the smoothness indicators that are readily available during computation. It is shown that this adaptive nature allows the resultant mapped WENO scheme to maintain sub-stencil weights close to the optimal weights in smooth regions without amplifying the weights of non-smooth stencils containing discontinuities. Therefore, adaptive mapping achieves enhanced accuracy in smooth regions and is more resistant against spurious oscillations near discontinuities. Taylor series analysis of the seventh order finite volume WENO scheme has been performed to demonstrate the loss of accuracy of the original WENO method near critical points. The convergence rates of the seventh order finite volume WENO scheme with adaptive mapping have been shown through a simple numerical example. Excellent results have been obtained for one-dimensional linear advection cases especially over long output times. Improved results have also been obtained for one- and two-dimensional Euler equation test cases.

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes.

High order finite-volume WENO scheme for five-equation model of compressible two-fluid flow

High order finite-volume weighted essentially non-oscillatory (WENO) scheme is applied for solving interface-capturing five-equation model of compressible two-fluid flows in one and two-space dimensions. The model is non-conservative and the governing equations consist of three equations, namely a continuity equation, a momentum equation and an energy balance equation for the fluid mixture and the remaining two are mass and energy equations for one of the two fluids. In the last equation, the non-conservative differential source term appears which is responsible for the energy exchange between fluids. The energy exchange is only due to mechanical work. The presence of non-conservative differential source terms in the two-fluid flow model introduce difficulties in developing high order accurate numerical schemes. The proposed numerical scheme is capable to preserve non-oscillatory property near strong discontinuities and gives high order accuracy in smooth regions. Different one and two-dimensional test problems are considered to analyze efficiency and accuracy of the proposed numerical algorithms. For validation, the solutions of proposed numerical scheme are compared with the results of already available high order kinetic flux-vector splitting scheme and discontinuous Galerkin scheme.

A new type of multi-resolution WENO schemes with increasingly higher order of accuracy

In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in [25,45], could obtain the optimal order of accuracy in smooth regions, and could simultaneously suppress spurious oscillations near discontinuities. The linear weights of such WENO schemes can be any positive numbers on the condition that their sum equals one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference and finite volume WENO schemes. These new WENO schemes are simple to construct and can be easily implemented to arbitrary high order of accuracy and in higher dimensions. Benchmark examples are given to demonstrate the robustness and good performance of these new WENO schemes.

Numerical study on the convergence to steady-state solutions of a new class of finite volume WENO schemes: triangular meshes

In this paper, we continue our research on the numerical study of convergence to steady-state solutions for a new class of finite volume weighted essentially non-oscillatory (WENO) schemes in Zhu and Shu (J Comput Phys 349:80–96, 2017), from tensor product meshes to triangular meshes. For the case of triangular meshes, this new class of finite volume WENO schemes was designed for time-dependent conservation laws in Zhu and Qiu (SIAM J Sci Comput 40(2):A903–A928, 2018) for the third- and fourth-order versions. In this paper, we extend the design to a new fifth-order version in the same framework to keep the essentially non-oscillatory property near discontinuities. Similar to the case of tensor product meshes in Zhu and Shu (2017), by performing such spatial reconstruction procedures together with a TVD Runge–Kutta time discretization, these WENO schemes do not suffer from slight post-shock oscillations that are responsible for the phenomenon wherein the residues of classical WENO schemes hang at a truncation error level instead of converging to machine zero. The third-, fourth-, and fifth-order finite volume WENO schemes in this paper can suppress the slight post-shock oscillations and have their residues settling down to a tiny number close to machine zero in steady-state simulations in our extensive numerical experiments.

New Finite Volume Weighted Essentially Nonoscillatory Schemes on Triangular Meshes

In this paper, we design a new type of high order finite volume weighted essentially nonoscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. The main advantages of these schemes are their compactness and robustness and that they could maintain a good convergence property for some steady state problems. Compared with the classical finite volume WENO schemes [C. Hu and C.-W. Shu, J. Comput. Phys., 150 (1999), pp. 97--127], the optimal linear weights are independent of the topological structure of the triangular meshes and can be any positive numbers with the one requirement that their summation is one. This is the first time any high order accuracy with the usage of only five unequal sized stencils in a spatial reconstruction methodology on triangular meshes has been obtained. Extensive numerical results are provided to illustrate the good performance of such new finite volume WENO schemes.

A High Order Well-Balanced Finite Volume WENO Scheme for a Blood Flow Model in Arteries

AbstractThe numerical simulations for the blood flow in arteries by high order accurate schemes have a wide range of applications in medical engineering. The blood flow model admits the steady state solutions, in which the flux gradient is non-zero and is exactly balanced by the source term. In this paper, we present a high order finite volume weighted essentially non-oscillatory (WENO) scheme, which preserves the steady state solutions and maintains genuine high order accuracy for general solutions. The well-balanced property is obtained by a novel source term reformulation and discretisation, combined with well-balanced numerical fluxes. Extensive numerical experiments are carried out to verify well-balanced property, high order accuracy, as well as good resolution for smooth and discontinuous solutions.

A New Type of Finite Volume WENO Schemes for Hyperbolic Conservation Laws

A new type of finite difference weighted essentially non-oscillatory (WENO) schemes for hyperbolic conservation laws was designed in Zhu and Qiu (J Comput Phys 318:110–121, 2016), in this continuing paper, we extend such methods to finite volume version in multi-dimensions. There are two major advantages of the new WENO schemes superior to the classical finite volume WENO schemes (Shu, in: Quarteroni (ed) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, CIME subseries, Springer, Berlin, 1998), the first is the associated linear weights can be any positive numbers with only requirement that their summation equals one, and the second is their simplicity and easy extension to multi-dimensions in engineering applications. The new WENO reconstruction is a convex combination of a fourth degree polynomial with two linear polynomials defined on unequal size spatial stencils in a traditional WENO fashion. These new fifth order WENO schemes use the same number of cell average information as the classical fifth order WENO schemes Shu (1998), could get less absolute numerical errors than the classical same order WENO schemes, and compress nonphysical oscillations nearby strong shocks or contact discontinuities. Some benchmark tests are performed to illustrate the capability of these schemes.

High order finite volume WENO schemes for the Euler equations under gravitational fields

Euler equations with gravitational source terms are used to model many astrophysical and atmospheric phenomena. This system admits hydrostatic balance where the flux produced by the pressure is exactly canceled by the gravitational source term, and two commonly seen equilibria are the isothermal and polytropic hydrostatic solutions. Exact preservation of these equilibria is desirable as many practical problems are small perturbations of such balance. High order finite difference weighted essentially non-oscillatory (WENO) schemes have been proposed in [22], but only for the isothermal equilibrium state. In this paper, we design high order well-balanced finite volume WENO schemes, which can preserve not only the isothermal equilibrium but also the polytropic hydrostatic balance state exactly, and maintain genuine high order accuracy for general solutions. The well-balanced property is obtained by novel source term reformulation and discretization, combined with well-balanced numerical fluxes. Extensive one- and two-dimensional simulations are performed to verify well-balanced property, high order accuracy, as well as good resolution for smooth and discontinuous solutions.