Weighted Essentially Non-oscillatory Reconstruction Research Articles

Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes I: Steady flows

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows.

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*

Three-dimensional high-order finite-volume method based on compact WENO reconstruction with hybrid unstructured grids

In the present work, a three-dimensional (3D) high-order finite-volume method, based on the compact simple weighted essentially non-oscillatory (WENO) reconstruction for hybrid unstructured grids, is developed. For each control cell, its stencils only involve neighboring cells. By ensuring that the various orders of the derivatives of the conservative variables are conserved over the stencils, an over-determined system of equations is formed, and the polynomials corresponding to each stencil are obtained by solving this system of equations using the least-squares method. The artificially determined weight coefficients, and the smoothing factors based on the smoothness of the flow field, are used to determine the nonlinear weights, which nonlinearly combine the polynomials of each stencil to obtain the final high-order reconstruction polynomial. Consequently, the conservative variables on the interface can be interpolated and the sphere-function based discrete gas-kinetic scheme (DGKS) is used to locally solve the Boltzmann equation. Lastly, the fluxes are calculated using the mesoscopic perspective based on the moment relations. The present algorithm has the following advantages: the stencils only involve the neighboring cells of each control volume, making the algorithm compact and simpler, especially for the operation of the boundary cells; the unknown coefficients are solved by the least-squares method to avoid the coefficient matrix being ill-conditioned; the simple WENO scheme artificially chooses the value to guarantee positive weights, in addition, the algorithm can be applied to 3D hybrid unstructured grids, making it more adaptable. In terms of flux calculation, DGKS not only inherits the advantages of the traditional gas-kinetic schemes (GKS) but also greatly simplifies the flux calculation expressions. Several cases are simulated to verify the correctness and robustness of the algorithm, and the accuracy test shows the algorithm is third-order accurate.

A sixth-order central WENO scheme for nonlinear degenerate parabolic equations

In this paper, we develop a new sixth-order finite difference central weighted essentially non-oscillatory (WENO) scheme with Z-type nonlinear weights for nonlinear degenerate parabolic equations. The centered polynomial is introduced for the WENO reconstruction to avoid the negative linear weights. We choose the Z-type nonlinear weights based on the $$L^2$$ -norm smoothness indicators, yielding the new WENO scheme with less computational cost. It is also confirmed that the proposed central WENO scheme with the devised nonlinear weights achieves sixth order accuracy in smooth regions. One- and two-dimensional numerical examples are presented to demonstrate the improved performance of the proposed central WENO scheme.

Dissipation-based WENO stabilization of high-order finite element methods for scalar conservation laws

We present a new perspective on the use of weighted essentially nonoscillatory (WENO) reconstructions in high-order methods for scalar hyperbolic conservation laws. The main focus of this work is on nonlinear stabilization of continuous Galerkin (CG) approximations. The proposed methodology also provides an interesting alternative to WENO-based limiters for discontinuous Galerkin (DG) methods. Unlike Runge–Kutta DG schemes that overwrite finite element solutions with WENO reconstructions, our approach uses a reconstruction-based smoothness sensor to blend the numerical viscosity operators of high- and low-order stabilization terms. The so-defined WENO approximation introduces low-order nonlinear diffusion in the vicinity of shocks, while preserving the high-order accuracy of a linearly stable baseline discretization in regions where the exact solution is sufficiently smooth. The underlying reconstruction procedure performs Hermite interpolation on stencils consisting of a mesh cell and its neighbors. The amount of numerical dissipation depends on the relative differences between partial derivatives of reconstructed candidate polynomials and those of the underlying finite element approximation. All derivatives are taken into account by the employed smoothness sensor. To assess the accuracy of our CG-WENO scheme, we derive error estimates and perform numerical experiments. In particular, we prove that the consistency error of the nonlinear stabilization is of the order p+1/2, where p is the polynomial degree. This estimate is optimal for general meshes. For uniform meshes and smooth exact solutions, the experimentally observed rate of convergence is as high as p+1.

Three-dimensional third-order gas-kinetic scheme on hybrid unstructured meshes for Euler and Navier–Stokes equations

In this paper, a third-order gas-kinetic scheme is developed on the three-dimensional hybrid unstructured meshes for the compressible inviscid and viscous flows. In the classical weighted essentially non-oscillatory (WENO) scheme, the high-order spatial accuracy is achieved by the non-linear combination of lower order polynomials. However, for the hybrid unstructured meshes, the procedures, including the selection of candidate stencils and calculation of linear weights, become extremely complicated, especially for three-dimensional problems. To overcome the drawbacks, a third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A general strategy is given for the selection of big stencil and sub-stencils for hybrid meshes, and the topologically independent linear weights are used in the spatial reconstruction. A unified interpolation is also used for the volume integral of different types of control volumes, as well as the flux integration over different cell interfaces. Incorporate with the two-stage fourth-order temporal discretization, the explicit high-order gas-kinetic schemes are developed for unsteady problems. With the lower–upper symmetric Gauss–Seidel (LU-SGS) methods and Jacobi iteration, the implicit high-order gas-kinetic schemes are developed for steady problems. To accelerate the computation, both explicit and implicit scheme is implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). Especially, for implicit parallel computation, Jacobi iteration is used. The speedup of GPU code suggests the potential of high-order gas-kinetic schemes for the large scale computation. A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy and robustness for both inviscid and viscous flows.

A sufficient condition for free-stream preserving WENO schemes on curvilinear grids of complex geometries

The well developed conservative finite-difference-method based on upwind schemes, such as the weighted essentially nonoscillatory (WENO) schemes, may violate the free-stream preserving (FP) property when applied to curvilinear grids due to the loss of the geometric conservation law. Recent developments in FP techniques have disregarded the balance between the computational accuracy and efficiency, and more importantly, the general regularity for the FP property is not available. In this paper, we rigorously prove that the upwind dissipation satisfies the FP property as long as the sub-stencil reconstructions of the metrics are identical by rearranging the dissipation part as an accumulative form composed of the sub-stencil reconstructions. With this, the novel FP metrics are constructed straightforwardly for various WENO reconstructions, introducing less complexity to implement the schemes with a merely simple accuracy compensation by replacing the central fluxes with high-order ones. Applications of such an implementation to the isotropic vortex convection and double Mach problem etc. validate the accuracy and robustness of the novel FP method in simulating both the smooth flows and discontinuities at lower computational costs than the previous methods. Besides, further viscous computations for the three-element airfoil and the 3D wing are conducted to show the good capability of the present method on complex geometries.

An arbitrary Lagrangian–Eulerian discontinuous Galerkin method for two‐dimensional compressible flows on adaptive quadrilateral meshes

AbstractIn this article, a discontinuous Galerkin (DG) scheme on the adaptive quadrilateral meshes is proposed to simulate two‐dimensional compressible flows in the direct arbitrary Lagrangian–Eulerian (ALE) framework. In our scheme, the Euler equations are discretized in the reference element with the help of a bilinear map. A kind of Taylor expansion basis functions in the reference element is used to construct the interpolation polynomials of variables. We describe the property that the material derivatives of the basis functions used in the DG discretization are equal to zero, with which the scheme is simplified. Furthermore, the mesh velocity in our ALE framework is obtained by implementing the approach of mesh movement based on the variational principle from [Adaptive mesh methods for one‐ and two‐dimensional hyperbolic conservation laws. SIAM J Numer Anal. 2003;41:487–515]. This approach of mesh movement automatically concentrates the mesh nodes near the regions with large gradient values of the variables and can greatly improve the resolution of the solution near these regions. In addition, a WENO (weighted essentially non‐oscillatory) reconstruction helps our scheme remove the numerical oscillations. Some numerical examples are presented to demonstrate the accuracy, high resolution, and robustness of our scheme.

Investigations on a high order SPH scheme using WENO reconstruction

Theoretically, a 2nd order convergence can be reached with the Smoothed Particle Hydrodynamics (SPH) method. However, depending on the spatial disorder of particles, the order of convergence observed in practice can be lower than one. In this paper, a methodology is proposed for the reconstruction of high order numerical fluxes in Riemann-SPH formulations for weakly-compressible flows, so as to increase the global order of convergence of the scheme. This methodology is based on the use of one-dimensional Weighted Essentially Non-Oscillatory (WENO) reconstructions applied at each pair of interacting particles, and each 1D WENO stencil is completed using Moving-Least-Squares (MLS) reconstructions. It is shown that a 6th order convergence can be reached with the proposed SPH-WENO scheme. The gain in accuracy and convergence properties of this scheme is shown and discussed through its application to various one-dimensional and two-dimensional test cases. In particular, the influence of the number of neighbor particles and of the spatial particle disorder is studied. Finally, it is shown that the proposed high-order SPH scheme provides a better accuracy to CPU time ratio than usual Riemann-SPH schemes. The treatment of boundary conditions on rigid walls with the proposed scheme is also discussed.

A class of high-order improved fast weighted essentially non-oscillatory schemes for achieving optimal order at any critical points

The weighted essentially non-oscillatory (WENO) scheme is one of the most popular shock-capturing schemes, and constructing a more efficient and higher-order WENO scheme has always been an intention of optimization design. In the general WENO reconstruction framework, the smoothness indicator plays an important role in identifying whether the sub-stencils are in discontinuous or smooth regions. However, the classical smoothness indicator is the most expensive one in the whole reconstruction algorithm, and its computational complexity increases sharply with the improvement of the accuracy order. Therefore, a class of efficient and superior WENO schemes called improved fast WENO (IFWENO) are proposed based on the fast WENO (FWENO). To improve efficiency, the smoothness indicator of the IFWENO scheme is simplified from the traditional version, and the nonlinear weight calculation method is modified. The parameter ε is carefully designed to obtain the superior property that the accuracy of the spatial derivatives will not degrade at any order critical point in smooth regions. The reason for the instability occurring in the high-order FWENO is revealed, and the parameter p is likewise specifically selected to improve robustness at discontinuities. The excellent multi-scale resolution of the proposed IFWENO scheme is proven by theoretical analyses and numerical experiments. Through several typical examples, the consistently high accuracy and efficiency of the designed scheme in both smooth and discontinuous regions are verified.

A Discontinuous Galerkin Overset Scheme Using WENO Reconstruction and Subcells for Two-Dimensional Problems

A new scheme for communication between overset grids using subcells and weighted essentially non oscillatory (WENO) reconstruction for two-dimensional problems has been proposed. The effectiveness of this procedure is demonstrated using the discontinuous Galerkin method. This scheme uses a standard WENO reconstruction by dividing the immediate neighbors into subcells to find the degrees of freedom in cells near the overset interface. This also has the added advantage that it also works as a limiter if a discontinuity passes through the overset interface. Accuracy tests to demonstrate the maintenance of higher order are provided. Results containing shocks are also provided to demonstrate the limiter aspect of the data communication procedure.

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.

High order implicit finite difference schemes with a semi-implicit WENO reconstruction for nonlinear degenerate parabolic equations

In this paper, we develop implicit finite difference schemes with a semi-implicit weighted essentially non-oscillatory (WENO) reconstruction for nonlinear degenerate parabolic equations. Such degenerate equations would exist finite speed front propagation which is similar to discontinuous solutions for hyperbolic equations. We propose to use the finite difference WENO scheme with an implicit time discretization, to avoid the parabolic time step condition from an explicit time discretization, namely Δt=O(Δx2) where Δx is the mesh size. We further simplify the resulting nonlinear system by employing a semi-implicit WENO reconstruction, that is, nonlinear weights are treated explicitly while keeping linear reconstructions of derivatives implicitly. The novelty is that the semi-implicit WENO reconstruction maintains the original high order accuracy both in space and in time, as well as unconditional stability as a fully implicit scheme, but the scheme is much simpler in complexity than the fully implicit one. Numerical experiments are performed to show the high order accuracy, large time step conditions with better efficiency as compared to an explicit scheme, and good performances for capturing front propagation.

A physical-constraint-preserving finite volume WENO method for special relativistic hydrodynamics on unstructured meshes

This paper presents a highly robust third-order accurate finite volume weighted essentially non-oscillatory (WENO) method for special relativistic hydrodynamics on unstructured triangular meshes. We rigorously prove that the proposed method is physical-constraint-preserving (PCP), namely, always preserves the positivity of the pressure and the rest-mass density as well as the subluminal constraint on the fluid velocity. The method is built on a highly efficient compact WENO reconstruction on unstructured meshes, a simple PCP limiter, the provably PCP property of the Harten–Lax–van Leer flux, and third-order strong-stability-preserving time discretization. Due to the relativistic effects, the primitive variables (namely, the rest-mass density, velocity, and pressure) are highly nonlinear implicit functions in terms of the conservative variables, making the design and analysis of our method nontrivial. To address the difficulties arising from the strong nonlinearity, we adopt a novel quasilinear technique for the theoretical proof of the PCP property. Three provable convergence-guaranteed iterative algorithms are also introduced for the robust recovery of primitive quantities from admissible conservative variables. We also propose a slight modification to an existing WENO reconstruction to ensure the scaling invariance of the nonlinear weights and thus to accommodate the homogeneity of the evolution operator, leading to the advantages of the modified WENO reconstruction in resolving multi-scale wave structures. Extensive numerical examples are presented to demonstrate the robustness, expected accuracy, and high resolution of the proposed method.

High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations

In this paper, we develop two finite difference weighted essentially non-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving the Degasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, which contain nonlinear high order derivatives, and possibly peakon solutions or shock waves. By introducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-elliptic system, and the \mdp equation as a first order system. Then we choose a linear finite difference scheme with suitable order of accuracy for the auxiliary variable(s), and two finite difference WENO schemes with unequal-sized sub-stencils for the primal variable. One WENO scheme uses one large stencil and several smaller stencils, and the other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENO scheme which uses several small stencils of the same size to make up a big stencil, both WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil and enjoy the freedom of arbitrary positive linear weights. Another advantage is that the final reconstructed polynomial on the target cell is a polynomial of the same degree as the polynomial over the big stencil, while the classical finite difference WENO reconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatory properties of the proposed schemes.

A hybrid WENO scheme for steady-state simulations of Euler equations

For strong shock waves in solutions of steady-state Euler equations, the high-order shock capturing schemes usually suffer from the difficulty of convergence of residue close to machine zero. Several new weighted essentially non-oscillatory (WENO) type schemes have recently been designed to overcome this long-standing issue. In this paper, a new hybrid strategy is proposed for the fifth-order WENO scheme to simulate steady-state solutions of Euler equations. Compared with the existing WENO schemes, the hybrid WENO scheme performs better steady-state convergence property with less dissipative and dispersive errors. Meanwhile, the essentially oscillation-free feature is kept. In the hybrid strategy, the stencil is distinguished into smooth, non-smooth, or transition regions, which is realized by a simple smoothness detector based on the smoothness indicators in the original WENO method. The linear reconstruction and the specific WENO reconstruction are applied to the smooth and non-smooth regions, respectively. In the transition region, the mixture of the linear and WENO reconstructions is adopted by a smooth transitive interpolation, which plays a vital role in the steady-state convergence for the hybrid scheme. Numerical comparisons and spectral analysis are presented to demonstrate the robust performance of the new hybrid scheme for steady-state Euler equations.

A high order positivity-preserving conservative WENO remapping method on 3D tetrahedral meshes

We propose a high order positivity-preserving conservative remapping method on three-dimensional (3D) tetrahedral meshes, based on the weighted essentially non-oscillatory (WENO) reconstruction method. By precisely computing the overlaps between the meshes before and after the rezoning step in the arbitrary Lagrangian–Eulerian (ALE) framework, our method does not limit the range of mesh movements and has wider applications. This also makes our remapping process simpler to attain high-order accuracy. We use the third order multi-resolution WENO reconstruction procedure in this paper as an example, in which we reconstruct three polynomials of different orders via nested central spatial stencils and distribute nonlinear weights based on the smoothness of the polynomials, ensuring optimal accuracy in the smooth region while avoiding numerical oscillations in the non-smooth region. The multi-resolution WENO procedure involves fewer high-order reconstruction polynomials and can use arbitrary positive linear weights, making it more effective for our 3D remapping problem. We incorporate an efficient local limiting to preserve positivity for the positive physical variables involved in the ALE framework without sacrificing the original high-order accuracy and conservation. A set of numerical tests are provided to verify properties of our remapping algorithm, such as high-order accuracy, conservation, essentially non-oscillatory performance, positivity-preserving and efficiency.

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.

Multi-level WENO schemes with an adaptive characteristic-wise reconstruction for system of Euler equations

The Weighted Essentially Non-Oscillatory (WENO) scheme is an accurate and robust reconstruction procedure to simulate compressible flows, especially in the presence of discontinuities in the solution. The recently introduced multi-level WENO schemes like WENO-ZQ (Zhu & Qiu, 2016), WENO-AO (Balsara et al. 2016), WENO-ZS (Zhu & Shu 2019) provide better resolution of the solution near discontinuities and accuracy for smooth problems than classical WENO scheme. Despite having better accuracy, a multi-level WENO scheme is not computationally efficient as it involves the computation of an extra smoothness indicator as compared to the classical WENO scheme. In this article, we have proposed two new algorithms of multi-level schemes for system of Euler equations, which retains the accuracy and resolution of multi-level schemes while being less costly. The first algorithm involves identifying the troubled-cells and employing the characteristic-wise WENO reconstruction in the troubled-cells to avoid spurious oscillations and component-wise WENO reconstruction in smooth regions. While in the second algorithm, the latter part is replaced by the component-wise finite difference upwind reconstruction. The important aspect of the new algorithm is the troubled-cell indicator, which has been developed utilizing the information of the extra smoothness indicator of multi-level schemes. The present algorithms are independent of order of multi-level WENO schemes. To demonstrate the new idea, we have considered fifth order WENO-AO(5,3), WENO-ZQ, and WENO-ZS schemes. Several 1D & 2D numerical experiments are performed to demonstrate the efficiency of proposed algorithms and its performance is compared with multi-level schemes. The proposed algorithms are efficient and take 15%–75% less computational time than multi-level schemes while retaining the advantageous features of multi-level schemes.

A high‐order weighted essentially nonoscillatory scheme based on exponential polynomials for nonlinear degenerate parabolic equations

AbstractIn this research the numerical solution of nonlinear degenerate parabolic equations is investigated by a new sixth‐order finite difference weighted essentially nonoscillatory (WENO) based on exponential polynomials. In smooth regions, the new scheme, named as EPWENO6, can achieve the maximal approximation order while in critical points it does not lose its accuracy. In order to better approximation near steep gradients without spurious oscillations, the EPWENO6 scheme is designed by the exponential polynomials that are incorporated into the WENO reconstruction. In design of nonlinear weights that play a very important role in WENO reconstructions, a global smoothness indicator using generalized undivided differences is introduced. To analyze the convergence order of the EPWENO6 method in full detail, the Lagrange‐type exponential functions have been used. By comparing the EPWENO6 scheme and other existing WENO schemes for solving degenerate parabolic equations, it can be seen that the new scheme offers better results while its computational cost is less. To illustrate the effectiveness of the proposed scheme, a number of numerical experiments are considered.