Finite Difference Weighted Essentially Non-oscillatory Research Articles

High-order finite difference scheme for compressible multi-component flow computations

This paper presents a high-order weighted essentially non-oscillatory (WENO) finite difference scheme for compressible multi-fluid and multi-phase dynamics. The scheme overcomes the difficulty of applying the common flux-based WENO finite difference scheme to multi-fluid problems by applying the reconstruction on primitive variables and, thus avoiding the spurious oscillations inherent in the standard methods. Schemes of orders up to nine are introduced and analyzed. The proposed finite difference schemes are significantly more efficient than the available high-order finite volume schemes in both storage requirement, operation counts, and inter-processor message passing in parallel computations with efficiency gains being higher at higher orders and higher spatial dimensions. For the same level of accuracy, in three-dimensional calculations, a fourfold speedup or higher at fifth-order accuracy or higher over the finite volume scheme is expected. A comparison of the proposed scheme with the standard flux-based finite difference scheme in solving a single-fluid shock small entropy wave interaction is presented, demonstrating its excellent performance. In a challenging, two-fluid problem of a strong shock interacting with a helium-air interface, the accuracy, non-oscillatory, conservation, and convergence of the scheme are illustrated. Moreover, in a liquid water-air shock tube problem, the scheme effectively captures compression and expansion waves and contact discontinuity, and outperforms low-order schemes. In computations of a two-dimensional shock bubble interaction, good agreements with experimental data are obtained and competitive performance to the high-order finite volume scheme is shown.

Numerical simulation of shock-disturbances interaction in high-speed compressible inviscid flow over a blunt nose using weighted essentially non-oscillatory scheme

In the present study, shock-disturbances interaction in high-speed compressible inviscid flow is simulated utilizing the weighted essentially non-oscillatory (WENO) scheme by employing the shock-capturing technique. For this aim, the two-dimensional Euler equations in strong conservative form are discretized by using the explicit third-order TVD Runge–Kutta scheme in time and the fifth-order WENO finite difference scheme in space. The main advantage of using the WENO scheme is its capability for properly solving the discontinuities in the domain without needing any artificial viscosity, limiter function or filter. Hence, this scheme is stable, and thus, it is suitable for simulating very high-speed flows with strong shocks. Three numerical tests are presented to demonstrate the remarkable capability of the WENO scheme and the corresponding numerical results are compared with the available analytical and numerical studies in the literature. The final and main test case is the steady and unsteady numerical simulation of the high-speed compressible inviscid flow past a two-dimensional circular cylinder. For this test case, firstly, the steady numerical solution at Mach=8.03 is provided. Then, the disturbances field is added to the upstream flow field and the unsteady numerical solution is performed for the fast and slow acoustic and also entropy waves. A sensitivity study is also performed to check the effects of the grid size and the type of flux splitting method on the solution. Both the steady and unsteady numerical solutions are compared with the existing analytical and numerical solutions and some conclusions are made.

A new type of finite difference WENO schemes for Hamilton–Jacobi equations

In this paper, we propose a new type of finite difference weighted essentially nonoscillatory (WENO) schemes to approximate the viscosity solutions of the Hamilton–Jacobi equations. The new scheme has three properties: (1) the scheme is fifth-order accurate in smooth regions while keep sharp discontinuous transitions with no spurious oscillations near discontinuities; (2) the linear weights can be any positive numbers with the symmetry requirement and that their sum equals one; (3) the scheme can avoid the clipping of extrema. Extensive numerical examples are provided to demonstrate the accuracy and the robustness of the proposed scheme.

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.

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.

Preconditioned WENO finite-difference lattice Boltzmann method for simulation of incompressible turbulent flows

In this work, a preconditioned high-order weighted essentially non-oscillatory (WENO) finite-difference lattice Boltzmann method (WENO-LBM) is applied to deal with the incompressible turbulent flows. Two different turbulence models namely, the Spalart–Allmaras (SA) and k−ωSST models are used and applied in the solution method for this aim. The spatial derivatives of the two-dimensional (2D) preconditioned LB equation in the generalized curvilinear coordinates are discretized by using the fifth-order WENO finite-difference scheme and an implicit–explicit Runge–Kutta scheme is adopted for the time discretization. For the convective and diffusive terms of the turbulence transport equations, the third-order WENO and second-order central finite-difference schemes are used, respectively. The preconditioning technique along with the local time-stepping method are applied to the WENO-LBM to further accelerate the solution to the converged steady-state condition, and therefore, an accurate and efficient incompressible LB solver is provided for simulating turbulent flows. The accuracy and robustness of the proposed solution method are assessed by computing two test cases: the 2D turbulent flow over a flat plate at Re=1.0×107 and the 2D turbulent flow over a NACA0012 airfoil at Re=6.0×106 and different angles of attack. The present results are compared with the available numerical and experimental results which show excellent agreement. It is demonstrated that the present solution methodology based on the WENO-LBM provides more accurate results of the incompressible turbulent flows and it requires lower number of grid points compared to the traditional (low-order accurate) LB and Navier–Stokes solvers.

A sixth-order finite difference WENO scheme for Hamilton–Jacobi equations

ABSTRACTIn this paper, a sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed to approximate the viscosity solution of the Hamilton–Jacobi equations. This new WENO scheme has the same spatial nodes as the classical fifth-order WENO scheme proposed by Jiang and Peng [Weighted ENO schemes for Hamilton–Jacobi equations, SIAM. J. Sci. Comput. 21 (2000), pp. 2126–2143] but can be as high as sixth-order accurate in smooth region while keeping sharp discontinuous transitions with no spurious oscillations near discontinuities. Extensive numerical experiments in one- and two-dimensional cases are carried out to illustrate the capability of the proposed scheme.

Wavelet-based adaptation methodology combined with finite difference WENO to solve ideal magnetohydrodynamics

In this paper, we present an accurate and efficient wavelet-based adaptive weighted essentially non-oscillatory (WENO) scheme for hydrodynamics and ideal magnetohydrodynamics (MHD) equations arising from the hyperbolic conservation systems. The proposed method works with the finite difference weighted essentially non-oscillatory (FD-WENO) method in space and the third order total variation diminishing (TVD) Runge–Kutta (RK) method in time. The philosophy of this work is to use the lifted interpolating wavelets as not only detector for singularities but also interpolator. Especially, flexible interpolations can be performed by an inverse wavelet transformation. When the divergence cleaning method introducing auxiliary scalar field ψ is applied to the base numerical schemes for imposing divergence-free condition to the magnetic field in a MHD equation, the approximations to derivatives of ψ require the neighboring points. Moreover, the fifth order WENO interpolation requires large stencil to reconstruct high order polynomial. In such cases, an efficient interpolation method is necessary. The adaptive spatial differentiation method is considered as well as the adaptation of grid resolutions. In order to avoid the heavy computation of FD-WENO, in the smooth regions fixed stencil approximation without computing the non-linear WENO weights is used, and the characteristic decomposition method is replaced by a component-wise approach. Numerical results demonstrate that with the adaptive method we are able to resolve the solutions that agree well with the solution of the corresponding fine grid.

Characteristic finite-difference WENO scheme for multicomponent compressible fluid analysis: Overestimated quasi-conservative formulation maintaining equilibriums of velocity, pressure, and temperature

The characteristic-interpolation-based finite-difference weighted essentially non-oscillatory (WENO) scheme, which maintains the equilibriums of velocity, pressure, and temperature, is implemented to simulate compressible multicomponent flow fields. We propose the overestimated quasi-conservative form of the characteristic-interpolation-based finite-difference WENO scheme. The proposed WENO scheme is written in the split form that has the consistent and dissipation parts of the numerical flux. The dissipation part of the numerical flux is in the conservative form to maintain the conservation of conservative variables. The scheme implemented in this study can maintain the equilibriums of velocity, pressure, and temperature in various one- and two-dimensional problems. The results of the present studies provide new insights into the vector form of numerical dissipation.

A Numerical Strategy for Freestream Preservation of the High Order Weighted Essentially Non-oscillatory Schemes on Stationary Curvilinear Grids

The weighted essentially non-oscillatory (WENO) schemes have been extensively employed for the simulation of complex flow fields due to their high order accuracy and good shock-capturing properties. However, the standard finite difference WENO scheme cannot hold freestream automatically in general curvilinear coordinates. Numerical errors from non-preserved freestream can hide small scales such as turbulent flow structures; aero-acoustic waves which can make the results inaccurate or even cause the simulation failure. To address this issue, a new numerical strategy to ensure freestream preservation properties of the WENO schemes on stationary curvilinear grids is proposed in this paper. The essential idea of this approach is to offset the geometrically induced errors by proper discretization of the metric invariants. It includes the following procedures: (1) the metric invariants are retained in the governing equations and the full forms of the transformed equations on the general curvilinear coordinates are solved; (2) the symmetrical, conservative form of the metrics instead of the original ones are used; (3) the WENO schemes which are applied for the inviscid fluxes of the governing equations are employed to compute the outer-level partial derivatives of the metric invariants. In other words, the outer-level derivative operators for the metric invariants are kept the same with those for the corresponding inviscid fluxes. It is verified theoretically in this paper that by using this approach, the WENO schemes hold the freestream preservation properties naturally and thus work well in the generalized coordinate systems. For some well-known WENO schemes, the derivative operators for the metric invariants are explicitly expressed and thus this approach can be straightforwardly employed. The effectiveness of this strategy is validated by several benchmark test cases.

Well-balanced finite difference WENO schemes for the Ripa model

The Ripa model is a non-homogeneous hyperbolic system with a non-zero source term, which is deduced from the shallow water equations by incorporating the horizontal temperature gradients. The Ripa model can maintain steady state solutions in which the flux gradients are non-zero but exactly balanced by the source term. It is a challenging task to design genuinely high order accurate numerical schemes which preserve exactly these steady state solutions. In this paper we design high order well-balanced finite difference weighted essentially non-oscillatory (WENO) schemes to this model. Rigorous theoretical analysis as well as extensive 1D and 2D numerical results all demonstrates that the present schemes share high order accuracy, maintain the well-balanced property, and keep good resolution for both smooth and discontinuous solutions.

A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws

In this paper a new simple fifth order weighted essentially non-oscillatory (WENO) scheme is presented in the finite difference framework for solving the hyperbolic conservation laws. The new WENO scheme is a convex combination of a fourth degree polynomial with two linear polynomials in a traditional WENO fashion. This new fifth order WENO scheme uses the same five-point information as the classical fifth order WENO scheme 14,20, could get less absolute truncation errors in L 1 and L ∞ norms, and obtain the same accuracy order in smooth region containing complicated numerical solution structures simultaneously escaping nonphysical oscillations adjacent strong shocks or contact discontinuities. The associated linear weights are artificially set to be any random positive numbers with the only requirement that their sum equals one. New nonlinear weights are proposed for the purpose of sustaining the optimal fifth order accuracy. The new WENO scheme has advantages over the classical WENO scheme 14,20 in its simplicity and easy extension to higher dimensions. Some benchmark numerical tests are performed to illustrate the capability of this new fifth order WENO scheme.

Projection methods for incompressible flow problems with WENO finite difference schemes

Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary convection-diffusion equations 20. This paper explores the applicability of these schemes for the simulation of incompressible flows. To this end, WENO schemes are used in several non-incremental and incremental projection methods for the incompressible Navier-Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov-Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious.

Preventing numerical oscillations in the flux‐split based finite difference method for compressible flows with discontinuities, II

SummaryProblems in the characteristic‐wise flux‐split based finite difference method when compressible flows with contact discontinuities or material interfaces are computed were presented and analyzed. The current analysis showed the following: (i) Even with the local characteristic decomposition technique, numerical errors could be caused by point‐wise flux vector splitting (FVS) methods, such as the Steger–Warming FVS or the van Leer FVS. Therefore, the Lax–Friedrichs type FVS method is required. (ii) If the isobars of a material are vertical lines, the combination of using the local characteristic decomposition and the global Lax–Friedrichs FVS can avoid velocity and pressure oscillations of contact discontinuities in this material for weighted essentially non‐oscillatory (WENO) schemes. (iii) For problems with material interfaces, the quasi‐conservative approach can be realized using characteristic‐wise flux‐split based finite difference WENO schemes if nonlinear WENO schemes in genuinely nonlinear characteristic fields can be guaranteed to be the same and the decomposition equation representing material interfaces is discretized properly. Copyright © 2015 John Wiley & Sons, Ltd.

Parallel adaptive mesh refinement method based on WENO finite difference scheme for the simulation of multi-dimensional detonation

For numerical simulation of detonation, computational cost using uniform meshes is large due to the vast separation in both time and space scales. Adaptive mesh refinement (AMR) is advantageous for problems with vastly different scales. This paper aims to propose an AMR method with high order accuracy for numerical investigation of multi-dimensional detonation. A well-designed AMR method based on finite difference weighted essentially non-oscillatory (WENO) scheme, named as AMR&WENO is proposed. A new cell-based data structure is used to organize the adaptive meshes. The new data structure makes it possible for cells to communicate with each other quickly and easily. In order to develop an AMR method with high order accuracy, high order prolongations in both space and time are utilized in the data prolongation procedure. Based on the message passing interface (MPI) platform, we have developed a workload balancing parallel AMR&WENO code using the Hilbert space-filling curve algorithm. Our numerical experiments with detonation simulations indicate that the AMR&WENO is accurate and has a high resolution. Moreover, we evaluate and compare the performance of the uniform mesh WENO scheme and the parallel AMR&WENO method. The comparison results provide us further insight into the high performance of the parallel AMR&WENO method.

A well-balanced finite difference WENO scheme for shallow water flow model

In this paper, we are concerned with shallow water flow model over non-flat bottom topography by high-order schemes. Most of the numerical schemes in the literature are developed from the original mathematical model of the shallow water flow. The novel contribution of this study consists in designing a finite difference weighted essentially non-oscillatory (WENO) scheme based on the alternative formulation of the shallow water flow model, denoted as “pre-balanced’’ shallow water equations and introduced in Rogers et al. (2003) [23]. This formulation greatly simplifies the achievement of the well-balancing of the present scheme. Rigorous numerical analysis as well as extensive numerical results all verify that the current scheme preserves the exact conservation property. It is important to note that this resulting scheme also maintains the non-oscillatory property near discontinuities and keeps high-order accuracy for smooth solutions at the same time.

Positivity-Preserving Finite Difference Weighted ENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations

In this paper, we utilize the maximum-principle-preserving flux limiting technique, originally designed for high-order weighted essentially nonoscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high-order positivity-preserving finite difference WENO methods for the ideal magnetohydrodynamic equations. Our scheme, under the constrained transport framework, can achieve high-order accuracy, a discrete divergence-free condition, and positivity of the numerical solution simultaneously. Numerical examples in one, two, and three dimensions are provided to demonstrate the performance of the proposed method.

Finite Difference Hermite WENO Schemes for Hyperbolic Conservation Laws

In this paper, a class of weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one and two dimensional nonlinear hyperbolic conservation law systems is presented. The construction of HWENO schemes is based on a finite difference formulation, Hermite interpolation, and nonlinearly stable Runge---Kutta methods. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and used in the reconstruction, while only the function values are evolved and used in the original WENO schemes. Comparing with the original finite difference WENO schemes of Jiang and Shu (J Comput Phys 126:202---228, 1996), one major advantage of HWENO schemes is its compactness in the reconstruction. For example, five points are needed in the stencil for a fifth order WENO (WENO5) reconstruction, while only three points are needed for a fifth order HWENO (HWENO5) reconstruction. Some benchmark numerical experiments are presented to illustrate efficiency of HWENO schemes.

A Survey of High Order Schemes for the Shallow Water Equations

In this paper, we survey our recent work on designing high order positivity- preserving well-balanced finite difference and finite volume WENO (weighted essen- tially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate in smooth regions for general solutions, are essentially non-oscillatory for general solutions with discontinuities, and at the same time they preserve exactly the water at rest or the more general moving water steady state solutions. A simple positivity-preserving limiter, valid under suitable CFL condi- tion, has been introduced in one dimension and reformulated to two dimensions with triangular meshes, and we prove that the resulting schemes guarantee the positivity of the water depth.

Finite difference weighted essentially non-oscillatory schemes with constrained transport for ideal magnetohydrodynamics

In this work we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work is to use efficient high-order WENO spatial discretizations with high-order strong stability-preserving Runge–Kutta (SSP-RK) time-stepping schemes. Numerical results have shown that with such methods we are able to resolve solution structures that are only visible at much higher grid resolutions with lower-order schemes. The key challenge in applying such methods to ideal MHD is to control divergence errors in the magnetic field. We achieve this by augmenting the base scheme with a novel high-order constrained transport approach that updates the magnetic vector potential. The predicted magnetic field from the base scheme is replaced by a divergence-free magnetic field that is obtained from the curl of this magnetic potential. The non-conservative weakly hyperbolic system that the magnetic vector potential satisfies is solved using a version of FD-WENO developed for Hamilton–Jacobi equations. The resulting numerical method is endowed with several important properties: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as point values on the same mesh (i.e., there is no mesh staggering); (2) both the spatial and temporal orders of accuracy are fourth-order; (3) no spatial integration or multidimensional reconstructions are needed in any step; and (4) special limiters in the magnetic vector potential update are used to control unphysical oscillations in the magnetic field. Several 2D and 3D numerical examples are presented to verify the order of accuracy on smooth test problems and to show high-resolution on test problems that involve shocks.