Finite Difference WENO Research Articles

Jet schemes for advection problems

We present a systematic methodology to develop high order accurate numerical approaches for linear advection problems. These methods are based on evolving parts of the jet of the solution in time, and are thus called jet schemes. Through the tracking of characteristics and the use of suitable Hermite interpolations, high order is achieved in an optimally local fashion, i.e. the update for the data at any grid point uses information from a single grid cell only. We show that jet schemes can be interpreted as advect-and-project processes in function spaces, where the projection step minimizes a stability functional. Furthermore, this function space framework makes it possible to systematically inherit update rules for the higher derivatives from the ODE solver for the characteristics. Jet schemes of orders up to five are applied in numerical benchmark tests, and systematically compared with classical WENO finite difference schemes. It is observed that jet schemes tend to possess a higher accuracy than WENO schemes of the same order.

Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation

AbstractIn this paper, we propose a new conservative semi-Lagrangian (SL) finite difference (FD) WENO scheme for linear advection equations, which can serve as a base scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedure in the proposed SL FD scheme is the same as the one used in the SL finite volume (FV) WENO scheme [3]. However, instead of inputting cell averages and approximate the integral form of the equation in a FV scheme, we input point values and approximate the differential form of equation in a FD spirit, yet retaining very high order (fifth order in our experiment) spatial accuracy. The advantage of using point values, rather than cell averages, is to avoid the second order spatial error, due to the shearing in velocity (v) and electrical field (E) over a cell when performing the Strang splitting to the Vlasov equation. As a result, the proposed scheme has very high spatial accuracy, compared with second order spatial accuracy for Strang split SL FV scheme for solving the Vlasov-Poisson (VP) system. We perform numerical experiments on linear advection, rigid body rotation problem; and on the Landau damping and two-stream instabilities by solving the VP system. For comparison, we also apply (1) the conservative SL FD WENO scheme, proposed in [22] for incompressible advection problem, (2) the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservative version of the SL FD WENO scheme in [3] to the same test problems. The performances of different schemes are compared by the error table, solution resolution of sharp interface, and by tracking the conservation of physical norms, energies and entropies, which should be physically preserved.

A Conservative Modification to the Ghost Fluid Method for Compressible Multiphase Flows

AbstractA conservative modification to the ghost fluid method (GFM) is developed for compressible multiphase flows. The motivation is to eliminate or reduce the conservation error of the GFM without affecting its performance. We track the conservative variables near the material interface and use this information to modify the numerical solution for an interfacing cell when the interface has passed the cell. The modification procedure can be used on the GFM with any base schemes. In this paper we use the fifth order finite difference WENO scheme for the spatial discretization and the third order TVD Runge-Kutta method for the time discretization. The level set method is used to capture the interface. Numerical experiments show that the method is at least mass and momentum conservative and is in general comparable in numerical resolution with the original GFM.

High order finite difference methods with subcell resolution for advection equations with stiff source terms

A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with the computed flow variables in the shock region modified by the Harten subcell resolution idea. For numerical experiments, a fifth-order finite-difference WENO scheme and its anti-diffusion WENO variant are considered. A wide range of 1D and 2D scalar and Euler system test cases are investigated. Studies indicate that for the considered test cases, the new method maintains high order accuracy in space for smooth flows, and for stiff source terms with discontinuities, it can capture the correct propagation speed of discontinuities in very coarse meshes with reasonable CFL numbers.

Simulations of Shallow Water Equations by Finite Difference WENO Schemes with Multilevel Time Discretization

Simulations of Shallow Water Equations by Finite Difference WENO Schemes with Multilevel Time Discretization

Relaxation method for unsteady convection–diffusion equations

We propose and implement a relaxation method for solving unsteady linear and nonlinear convection–diffusion equations with continuous or discontinuity-like initial conditions. The method transforms a convection–diffusion equation into a relaxation system, which contains a stiff source term. The resulting relaxation system is then solved by a third-order accurate implicit–explicit (IMEX) Runge–Kutta method in time and a fifth-order finite difference WENO scheme in space. Numerical results show that the method can be used to effectively solve convection–diffusion equations with both smooth structures and discontinuities.

High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws

In [10], the authors have designed a new fifth order WENO finite-difference scheme by adding a higher order smoothness indicator which is obtained as a simple and inexpensive linear combination of the already existing low order smoothness indicators. Moreover, this new scheme, dubbed as WENO-Z, has a CPU cost which is equivalent to the one of the classical WENO-JS [2], and smaller than that of the mapped WENO-M, [5], since it involves no mapping of the nonlinear weights. In this article, we take a closer look at Taylor expansions of the Lagrangian polynomials of the WENO substencils and the related inherited symmetries of the classical lower order smoothness indicators to obtain a general formula for the higher order smoothness indicators that allows the extension of the WENO-Z scheme to all (odd) orders of accuracy. We further investigate the improved accuracy of the WENO-Z schemes at critical points of smooth solutions as well as their distinct numerical features as a result of the new sets of nonlinear weights and we show that regarding the numerical dissipation WENO-Z occupies an intermediary position between WENO-JS and WENO-M. Some standard numerical experiments such as the one dimensional Riemann initial values problems for the Euler equations and the Mach 3 shock density-wave interaction and the two dimensional double-Mach shock reflection problems are presented.

A massively parallel multi-block hybrid compact–WENO scheme for compressible flows

A new multi-block hybrid compact–WENO finite-difference method for the massively parallel computation of compressible flows is presented. In contrast to earlier methods, our approach breaks the global dependence of compact methods by using explicit finite-difference methods at block interfaces and is fully conservative. The resulting method is fifth- and sixth-order accurate for the convective and diffusive fluxes, respectively. The impact of the explicit interface treatment on the stability and accuracy of the multi-block method is quantified for the advection and diffusion equations. Numerical errors increase slightly as the number of blocks is increased. It is also found that the maximum allowable time steps increase with the number of blocks. The method demonstrates excellent scalability on up to 1264 processors.

Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the d-function. For the approximation of the d-function, the direct projection method is used that was proposed in (6). The d-function is constructed in a consistent way to the derivative operator. Non- linear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The d-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singu- lar source, however, the solution of the linear scalar wave equation with the non- stationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.

Velocity gradient in Kelvin-Helmholtz instability for supersonic fluid

Two-dimensional numerical calculations using weighted essentially non-oscillatory schemes （WENO） scheme were performed to study velocity gradient in the Kelvin-Helmholtz （KH） instability for supersonic fluid. It is found that the velocity gradient has a stabilization effect on the KH instability for supersonic fluid, and the linear growth rate empirical formula with velocity gradient stabilization effect is deduced. The empirical formula with velocity gradient stabilization effect agrees well will the simulation results. The sharp density contour is obtained, which indicates that the WENO finite difference scheme has good capturing ability in interface deformation. The evolution of KH vortex is influenced by the velocity gradient. When the density gradient is fixed, it is found that the larger the velocity gradient the smaller the transverse scale of the KH vortex.

A hybrid scheme for solving a multi-class traffic flow model with complex wave breaking

This paper extends the δ -mapping algorithm to solve a multi-class traffic flow model on an inhomogeneous highway, which is characterized by spatially varying fluxes and very complex waves. In the algorithm, the values of solution variables are mapped onto an intermediate state locally around the cell interface, and then the finite difference WENO reconstruction is applied. The hybrid scheme is consistent with steady flows, and resolves the waves more efficiently than the standard WENO scheme. The algorithm is also applicable to hyperbolic conservation laws with spatially varying fluxes in general.

Hybrid finite compact‐WENO schemes for shock calculation

AbstractHybrid finite compact (FC)‐WENO schemes are proposed for shock calculations. The two sub‐schemes (finite compact difference scheme and WENO scheme) are hybridized by means of the similar treatment as in ENO schemes. The hybrid schemes have the advantages of FC and WENO schemes. One is that they possess the merit of the finite compact difference scheme, which requires only bi‐diagonal matrix inversion and can apply the known high‐resolution flux to obtain high‐performance numerical flux function; another is that they have the high‐resolution property of WENO scheme for shock capturing. The numerical results show that FC‐WENO schemes have better resolution properties than both FC‐ENO schemes and WENO schemes. In addition, some comparisons of FC‐ENO and artificial compression method (ACM) filter scheme of Yee et al. are also given. Copyright © 2006 John Wiley & Sons, Ltd.

High order Hybrid central—WENO finite difference scheme for conservation laws

In this article we present a high resolution hybrid central finite difference—WENO scheme for the solution of conservation laws, in particular, those related to shock–turbulence interaction problems. A sixth order central finite difference scheme is conjugated with a fifth order weighted essentially non-oscillatory WENO scheme in a grid-based adaptive way. High order multi-resolution analysis is used to detect the high gradients regions of the numerical solution in order to capture the shocks with the WENO scheme while the smooth regions are computed with the more efficient and accurate central finite difference scheme. The application of high order filtering to mitigate the dispersion error of central finite difference schemes is also discussed. Numerical experiments with the 1D compressible Euler equations are shown.

Efficient time integration of the Boltzmann-Poisson system applied to semiconductor device simulation

This paper presents new efficient methods for performing the time integration of finite-difference WENO approximations of the Boltzmann-Poisson system applied to semiconductor device simulation. The developed methods are based on local time-step schemes for hyperbolic conservation laws which permit the use of different time increments at different positions in space. A strategy to dynamically adapt the space-time grid according to the actual stability criteria imposed by the CFL-condition is proposed. The resulting numerical schemes are used to simulate the electron transport in a n+−n−n+ silicon diode and in a silicon MESFET. Several numerical tests and comparisons with computations performed with TVD Runge-Kutta type algorithms prove the efficiency of the presented time integration methods.

On different flux splittings and flux functions in WENO schemes for balance laws

In this paper we focus our attention on obtaining well-balanced schemes for balance laws by using Marquina’s flux in combination with the finite difference and finite volume WENO schemes. We consider also the Rusanov flux splitting and the HLL approximate Riemann solver. In particular, for the presented numerical schemes we develop corresponding discretizations of the source term, based on the idea of balancing with the flux gradient. When applied to the open-channel flow and to the shallow water equations, we obtain the finite difference WENO scheme with Marquina’s flux splitting, which satisfies the approximate conservation property, and also the balanced finite volume WENO scheme with Marquina’s solver satisfying the exact conservation property. Finally, we also present an improvement of the balanced finite difference WENO scheme with the Rusanov (locally Lax–Friedrichs) flux splitting, we previously developed in [Vuković S, Sopta L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J Comput Phys 2002;179:593–621].

High order finite difference WENO schemes with the exact conservation property for the shallow water equations

Shallow water equations with nonflat bottom have steady state solutions in which the flux gradients are nonzero but exactly balanced by the source term. It is a challenge to design genuinely high order accurate numerical schemes which preserve exactly these steady state solutions. In this paper we design high order finite difference WENO schemes to this system with such exact conservation property (C-property) and at the same time maintaining genuine high order accuracy. Extensive one and two dimensional simulations are performed to verify high order accuracy, the exact C-property, and good resolution for smooth and discontinuous solutions.

Anti-diffusive High Order WENO Schemes for Hamilton-Jacobi Equations

In this paper, we generalize the technique of anti-diffusive flux corrections for high order finite difference WENO schemes solving conservation laws in, to solve Hamilton-Jacobi equations. The objective is to obtain sharp resolution for kinks, which are derivative discontinuities in the viscosity solutions of Hamilton-Jacobi equations. We would like to resolve kinks better while maintaining high order accuracy in smooth regions. Numerical examples for one and two space dimensional problems demonstrate the good quality of these Hamiltonian corrected WENO schemes.

Multidomain WENO Finite Difference Method with Interpolation at Subdomain Interfaces

High order finite difference WENO methods have the advantage of simpler coding and smaller computational cost for multi-dimensional problems, compared with finite volume WENO methods of the same or...

A WENO-solver for the transients of Boltzmann–Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods

In this paper we develop a deterministic high order accurate finite-difference WENO solver to the solution of the 1-D Boltzmann–Poisson system for semiconductor devices. We follow the work in Fatemi and Odeh [9] and in Majorana and Pidatella [16] to formulate the Boltzmann–Poisson system in a spherical coordinate system using the energy as one of the coordinate variables, thus reducing the computational complexity to two dimensions in phase space and dramatically simplifying the evaluations of the collision terms. The solver is accurate in time hence potentially useful for time-dependent simulations, although in this paper we only test it for steady-state devices. The high order accuracy and nonoscillatory properties of the solver allow us to use very coarse meshes to get a satisfactory resolution, thus making it feasible to develop a 2-D solver (which will be five dimensional plus time when the phase space is discretized) on today’s computers. The computational results have been compared with those by a Monte Carlo simulation and excellent agreements have been found. The advantage of the current solver over a Monte Carlo solver includes its faster speed, noise-free resolution, and easiness for arbitrary moment evaluations. This solver is thus a useful benchmark to check on the physical validity of various hydrodynamic and energy transport models. Some comparisons have been included in this paper.