Weighted Essentially Non-oscillatory Method Research Articles

Operator splitting implicit integration factor methods for stiff reaction–diffusion–advection systems

For reaction–diffusion–advection equations, the stiffness from the reaction and diffusion terms often requires very restricted time step size, while the nonlinear advection term may lead to a sharp gradient in localized spatial regions. It is challenging to design numerical methods that can efficiently handle both difficulties. For reaction–diffusion systems with both stiff reaction and diffusion terms, implicit integration factor (IIF) method and its higher dimensional analog compact IIF (cIIF) serve as an efficient class of time-stepping methods, and their second order version is linearly unconditionally stable. For nonlinear hyperbolic equations, weighted essentially non-oscillatory (WENO) methods are a class of schemes with a uniformly high order of accuracy in smooth regions of the solution, which can also resolve the sharp gradient in an accurate and essentially non-oscillatory fashion. In this paper, we couple IIF/cIIF with WENO methods using the operator splitting approach to solve reaction–diffusion–advection equations. In particular, we apply the IIF/cIIF method to the stiff reaction and diffusion terms and the WENO method to the advection term in two different splitting sequences. Calculation of local truncation error and direct numerical simulations for both splitting approaches show the second order accuracy of the splitting method, and linear stability analysis and direct comparison with other approaches reveals excellent efficiency and stability properties. Applications of the splitting approach to two biological systems demonstrate that the overall method is accurate and efficient, and the splitting sequence consisting of two reaction–diffusion steps is more desirable than the one consisting of two advection steps, because CWC exhibits better accuracy and stability.

RKDG methods with WENO type limiters and conservative interfacial procedure for one-dimensional compressible multi-medium flow simulations

In this paper, we continue on studying the Runge–Kutta discontinuous Galerkin (RKDG) methods to solve compressible multi-medium flow with conservative treatment of the moving material interface. Comparing with the paper by J. Qiu, T.G. Liu and B.C. Khoo [J. Comput. Phys. 222 (2007) 353–373], we adopt the HLLC flux instead of Lax-Friedrichs numerical flux, the finite volume weighted essentially nonoscillatory (WENO) and Hermite WENO (HWENO) reconstructions as limiter instead of TVB limiter for RKDG. The HLLC flux is based on the approximate Riemann solver with little numerical viscosity and can resolve the contact discontinuity and shear wave very well. For limiter procedure, first we use the KXRCF indicator to identify the troubled cell, then apply WENO or HWENO method to reconstruct the polynomial in the troubled cell, while maintaining the cell average. This limiter procedure is more accurate and less problem dependent than the TVB limiter. Numerical results in one dimension for multi-medium flows such as gas–gas and gas–water are provided to illustrate the capability of these procedures.

Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow

In this paper, we propose a semi-Lagrangian finite difference formulation for approximating conservative form of advection equations with general variable coefficients. Compared with the traditional semi-Lagrangian finite difference schemes [5,25], which approximate the advective form of the equation via direct characteristics tracing, the scheme proposed in this paper approximates the conservative form of the equation. This essential difference makes the proposed scheme naturally conservative for equations with general variable coefficients. The proposed conservative semi-Lagrangian finite difference framework is coupled with high order essentially non-oscillatory (ENO) or weighted ENO (WENO) reconstructions to achieve high order accuracy in smooth parts of the solution and to capture sharp interfaces without introducing spurious oscillations. The scheme is extended to high dimensional problems by Strang splitting. The performance of the proposed schemes is demonstrated by linear advection, rigid body rotation, swirling deformation, and two dimensional incompressible flow simulation in the vorticity stream-function formulation. As the information is propagating along characteristics, the proposed scheme does not have the CFL time step restriction of the Eulerian method, allowing for a more efficient numerical realization for many application problems.

A fast and accurate solver for the general rate model of column liquid chromatography

Practical application of the general rate model is often hindered by computational complexity and by unknown parameter values. Model simplifications and parameter correlations are not always applicable, and repeated model solutions are required for parameter estimation and process optimization. We have hence developed and implemented a fast and accurate solver for the general rate model with various adsorption isotherms. Several state of the art scientific computing techniques have been combined for maximal solver performance: (1) the model equations are spatially discretized with finite volumes and the weighted essentially non-oscillatory (WENO) method. (2) A non-commercial solver with variable stepwidth and order is applied for time integration. (3) The internal linear solver module is replaced by customized code that is based on domain decomposition and can be executed on parallel computers. We demonstrate the speed and accuracy of our solver with numerical examples. The WENO method significantly improves accuracy even on rather coarse grids, and the solver runtime scales linearly with relevant grid sizes and processor numbers. Code parallelization is essential for utilizing the full power of multiprocessor and multicore technology in modern personal computers. The general rate model of a strongly non-linear two component system is solved in few seconds.

Multi-Phase Modeling for Water Waves over Submerged Structures Using WENO and Level Set Method

A multiphase model for water waves was developed consisting of the fifth order space accurate WENO (weighted essen-tially non-oscillatory) method for advection terms and a level set method for capturing the gas-liquid interface. The model was applied to a solitary wave propagation over a rectangular structure, 2nd-order Stokes waves over a trapezoidal struc-ture, and a solitary wave over a semicircular structure. Through comparisons with existing experiments, the model was found to be capable of reproducing the experimental results with relatively coarse grids and without an explicit turbulence model because of the adoptation of high-order and high resolution scheme with the concept of Implicit LES.

A conservative high order semi-Lagrangian WENO method for the Vlasov equation

In this paper, we propose a novel Vlasov solver based on a semi-Lagrangian method which combines Strang splitting in time with high order WENO (weighted essentially non-oscillatory) reconstruction in space. A key insight in this work is that the spatial interpolation matrices, used in the reconstruction process of a semi-Lagrangian approach to linear hyperbolic equations, can be factored into right and left flux matrices. It is the factoring of the interpolation matrices which makes it possible to apply the WENO methodology in the reconstruction used in the semi-Lagrangian update. The spatial WENO reconstruction developed for this method is conservative and updates point values of the solution. While the third, fifth, seventh and ninth order reconstructions are presented in this paper, the scheme can be extended to arbitrarily high order. WENO reconstruction is able to achieve high order accuracy in smooth parts of the solution while being able to capture sharp interfaces without introducing oscillations. Moreover, the CFL time step restriction of a regular finite difference or finite volume WENO scheme is removed in a semi-Lagrangian framework, allowing for a cheaper and more flexible numerical realization. The quality of the proposed method is demonstrated by applying the approach to basic test problems, such as linear advection and rigid body rotation, and to classical plasma problems, such as Landau damping and the two-stream instability. Even though the method is only second order accurate in time, our numerical results suggest the use of high order reconstruction is advantageous when considering the Vlasov–Poisson system.

High‐order simulation of polymorphic crystallization using weighted essentially nonoscillatory methods

AbstractMost pharmaceutical manufacturing processes include a series of crystallization processes to increase purity with the last crystallization used to produce crystals of desired size, shape, and crystal form. The fact that different crystal forms (known as polymorphs) can have vastly different characteristics has motivated efforts to understand, simulate, and control polymorphic crystallization processes. This article proposes the use of weighted essentially nonoscillatory (WENO) methods for the numerical simulation of population balance models (PBMs) for crystallization processes, which provide much higher order accuracy than previously considered methods for simulating PBMs, and also excellent accuracy for sharp or discontinuous distributions. Three different WENO methods are shown to provide substantial reductions in numerical diffusion or dispersion compared with the other finite difference and finite volume methods described in the literature for solving PBMs, in an application to the polymorphic crystallization of L‐glutamic acid. © 2008 American Institute of Chemical Engineers AIChE J, 2009

Implicit preconditioned WENO scheme for steady viscous flow computation

A class of lower–upper symmetric Gauss–Seidel implicit weighted essentially nonoscillatory (WENO) schemes is developed for solving the preconditioned Navier–Stokes equations of primitive variables with Spalart–Allmaras one-equation turbulence model. The numerical flux of the present preconditioned WENO schemes consists of a first-order part and high-order part. For first-order part, we adopt the preconditioned Roe scheme and for the high-order part, we employ preconditioned WENO methods. For comparison purpose, a preconditioned TVD scheme is also given and tested. A time-derivative preconditioning algorithm is devised and a discriminant is devised for adjusting the preconditioning parameters at low Mach numbers and turning off the preconditioning at intermediate or high Mach numbers. The computations are performed for the two-dimensional lid driven cavity flow, low subsonic viscous flow over S809 airfoil, three-dimensional low speed viscous flow over 6:1 prolate spheroid, transonic flow over ONERA-M6 wing and hypersonic flow over HB-2 model. The solutions of the present algorithms are in good agreement with the experimental data. The application of the preconditioned WENO schemes to viscous flows at all speeds not only enhances the accuracy and robustness of resolving shock and discontinuities for supersonic flows, but also improves the accuracy of low Mach number flow with complicated smooth solution structures.

A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids

A discontinuous Galerkin method based on a Taylor basis is presented for the solution of the compressible Euler equations on arbitrary grids. Unlike the traditional discontinuous Galerkin methods, where either standard Lagrange finite element or hierarchical node-based basis functions are used to represent numerical polynomial solutions in each element, this DG method represents the numerical polynomial solutions using a Taylor series expansion at the centroid of the cell. Consequently, this formulation is able to provide a unified framework, where both cell-centered and vertex-centered finite volume schemes can be viewed as special cases of this discontinuous Galerkin method by choosing reconstruction schemes to compute the derivatives, offer the insight why the DG methods are a better approach than the finite volume methods based on either TVD/MUSCL reconstruction or essentially non-oscillatory (ENO)/weighted essentially non-oscillatory (WENO) reconstruction, and has a number of distinct, desirable, and attractive features, which can be effectively used to address some of shortcomings of the DG methods. The developed method is used to compute a variety of both steady-state and time-accurate flow problems on arbitrary grids. The numerical results demonstrated the superior accuracy of this discontinuous Galerkin method in comparison with a second order finite volume method and a third-order WENO method, indicating its promise and potential to become not just a competitive but simply a superior approach than its finite volume and ENO/WENO counterparts for solving flow problems of scientific and industrial interest.

Numerical experiments with several variant WENO schemes for the Euler equations

AbstractNumerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126:202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.

Observations on the fifth-order WENO method with non-uniform meshes

The weighted essentially non-oscillatory (WENO) methods are a popular high-order spatial discretization for hyperbolic partial differential equations. Typical treatments of WENO methods assume a uniform mesh. In this paper, we give explicit formulas for the finite-volume, fifth-order WENO (WENO5) method on non-uniform meshes in a way that is amenable to efficient implementation. We then compare the performance of the non-uniform mesh approach with the classical uniform mesh approach for the finite-volume formulation of the WENO5 method. We find that the numerical results significantly favor the non-uniform mesh approach both in terms of computational efficiency as well as memory usage. We expect this investigation to provide a basis for future work on adaptive mesh methods coupled with the finite-volume WENO methods.

Stencil Adaptation Properties of a WENO Scheme in Direct Numerical Simulations of Compressible Turbulence

Weighted essentially non-oscillatory (WENO) methods can simultaneously provide the high order of accuracy, high bandwidth-resolving efficiency, and shock-capturing capability required for the detailed simulation of compressible turbulence. However, rigorous analysis of the actual versus theoretical error properties of these non-linear numerical methods is difficult. We use a bandwidth-optimized WENO scheme to conduct direct numerical simulations of two- and three-dimensional decaying isotropic turbulence, and we evaluate the performance of quantitative indicators of local WENO adaptation behavior within the resulting flow fields. One aspect of this assessment is the demarcation of shock-containing and smooth regions where the WENO method should, respectively, engage its adaptation mechanism and revert to its linear optimal stencil. Our results show that these indicators, when synthesized properly, can provide valuable quantitative information suitable for statistical characterization.

Linear Instability of the Fifth-Order WENO Method

The weighted essentially nonoscillatory (WENO) methods are popular spatial discretization methods for hyperbolic partial differential equations. In this paper we show that the combination of the widely used fifth-order WENO spatial discretization (WENO5) and the forward Euler time integration method is linearly unstable when numerically integrating hyperbolic conservation laws. Consequently it is not convergent. Furthermore we show that all two-stage, second-order explicit Runge–Kutta (ERK) methods are linearly unstable (and hence do not converge) when coupled with WENO5. We also show that all optimal first- and second-order strong-stability-preserving (SSP) ERK methods are linearly unstable when coupled with WENO5. Moreover the popular three-stage, third-order SSP(3,3) ERK method offers no linear stability advantage over non-SSP ERK methods, including ones with negative coefficients, when coupled with WENO5. We give new linear stability criteria for combinations of WENO5 with general ERK methods of any order. We find that a sufficient condition for the combination of an ERK method and WENO5 to be linearly stable is that the linear stability region of the ERK method should include the part of the imaginary axis of the form $[-\iota \mu,\iota \mu]$ for some $\mu>0$. The linear stability analysis also provides insight into the behavior of ERK methods applied to nonlinear problems and problems with discontinuous solutions. We confirm the assertions of our analysis by means of numerical tests.

Effects of WENO flux reconstruction order and spatial resolution on reshocked two-dimensional Richtmyer–Meshkov instability

Weighted essentially non-oscillatory (WENO) simulations of the reshocked two-dimensional single-mode Richtmyer–Meshkov instability using third-, fifth- and ninth-order spatial flux reconstruction and uniform grid resolutions corresponding to 128, 256 and 512 points per initial perturbation wavelength are presented. The dependence of the density, vorticity, simulated density Schlieren and baroclinic production fields, mixing layer width, circulation deposition, mixing profiles, production and mixing fractions, energy spectra, statistics, probability distribution functions, numerical turbulent kinetic energy and enstrophy production/dissipation rates, numerical Reynolds numbers, and numerical viscosity on the order and resolution is investigated to long evolution times. The results are interpreted using the implicit numerical dissipation in the characteristic projection-based, finite-difference WENO method. It is shown that higher-order higher-resolution simulations have lower numerical dissipation. The sensitivity of the quantities considered to the order and resolution is further amplified following reshock, when the energy deposition by the second shock-interface interaction induces the formation of small-scale structures. Lower-order lower-resolution simulations preserve large-scale structures and flow symmetry to late times, while higher-order higher-resolution simulations exhibit fragmentation of the structures, symmetry breaking and increased mixing. Similar flow features are qualitatively and quantitatively captured by either approximately doubling the order or the resolution. Additionally, the computational scaling shows that increasing the order is more advantageous than increasing the resolution for the flow considered here. The present investigation suggests that the ninth-order WENO method is well-suited for the simulation and analysis of complex multi-scale flows and mixing generated by shock-induced hydrodynamic instabilities.

A Fifth Order Flux Implicit WENO Method

The weighted essentially non-oscillatory (WENO) method is an excellent spatial discretization for hyperbolic partial differential equations with discontinuous solutions. However, the time-step restriction associated with explicit methods may pose severe limitations on their use in applications requiring large scale computations. An efficient implicit WENO method is necessary. In this paper, we propose a prototype flux-implicit WENO (iWENO) method. Numerical tests on classical scalar equations show that this is a viable and stable method, which requires appropriate time-stepping methods. Future study will include the examination of such methods as well as extension of iWENO to systems and higher dimensional problems.

A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill–Whitham–Richards traffic flow model

In this paper, we present a high-order weighted essentially non-oscillatory (WENO) scheme for solving a multi-class extension of the Lighthill–Whitham–Richards (LWR) model. We first review the multi-class LWR model and present some of its analytical properties. We then present the WENO schemes, which were originally designed for computational fluid dynamics problems and for solving hyperbolic conservation laws in general, and demonstrate how to apply these to the present model. We found through numerical experiments that the WENO method is vastly more efficient than the low-order Lax–Friedrichs scheme, yet both methods converge to the same solution of the physical model. It is especially interesting to observe the small staircases in the solution which are completely missed out, because of the numerical viscosity, if a lower-order method is used without a sufficiently refined mesh. To demonstrate the applicability of this new, efficient numerical tool, we study the multi-class model under different parameter regimes and traffic stream models. We consider also the convergence of the multi-class LWR model when the number of classes goes to infinity. We show that the solution converges to a smooth profile without staircases when the number of classes increases.

Discussion on: “An Optimal Control Approach for Hybrid Systems”

A finite volume approximation of the scalar hyperbolic conservation law or advection–diffusion equation is given. In the context of the method of lines, the space discretization uses weighted essentially non oscillatory (WENO) reconstructions with adaptive order (WENO-AO), and the time evolution uses implicit Runge–Kutta methods. Therefore the timestep may be larger than the CFL timestep. To reduce oscillation in the solution, ideas related to spatially partitioned Runge–Kutta methods are used. An adaptive Runge–Kutta method is developed that blends the L-stable, third order, implicit Radau IIA method with the composite backward Euler method using a weighting procedure inspired from spatial WENO methods. The weighting procedure requires a smoothness indicator, and several possibilities are considered, although one is perhaps seen to be preferred. The overall scheme is proven to maintain third order accuracy when the solution is smooth. When the solution has a discontinuity, the scheme is shown computationally to be third order accurate away from shocks, and to achieve the overall accuracy of the backward Euler method. Numerical examples show that the adaptive Runge–Kutta method reduces oscillations in the solution. Moreover, the resulting scheme is shown to be unconditionally L-stable for smooth solutions to the linear problem.

Numerical Comparison of WENO Finite Volume and Runge–Kutta Discontinuous Galerkin Methods

High order WENO (weighted essentially non-oscillatory) schemes and discontinuous Galerkin methods are two classes of high order, high resolution methods suitable for convection dominated simulations with possible discontinuous or sharp gradient solutions. In this paper we first review these two classes of methods, pointing out their similarities and differences in algorithm formulation, theoretical properties, implementation issues, applicability, and relative advantages. We then present some quantitative comparisons of the third order finite volume WENO methods and discontinuous Galerkin methods for a series of test problems to assess their relative merits in accuracy and CPU timing.

Third order WENO scheme on sparse grids for hyperbolic equations

The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). The computational cost of such schemes increases significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points and nonlinearity of high order accuracy WENO schemes. How to achieve fast computations by WENO methods for high spatial dimension PDEs is a challenging and important question. Recently, sparse-grid has become a major approximation tool for high dimensional problems. The open question is how to design WENO computations on sparse grids such that comparable high order accuracy of WENO schemes in smooth regions and essentially non-oscillatory stability in non-smooth regions of the solutions can still be achieved as that for computations on regular single grids? In this paper, we combine the third order finite difference WENO method with sparse-grid combination technique and solve high spatial dimension hyperbolic equations on sparse grids. WENO interpolation is proposed for the prolongation part in sparse grid combination techniques to deal with discontinuous solutions of hyperbolic equations. Numerical examples are presented to show that significant computational times are saved while both high order accuracy and stability of the WENO scheme are maintained for simulations on sparse grids.