High-order WENO Schemes Research Articles

A New Type of High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes

A New Type of High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes

Numerical solution of the Vlasov-Poisson equations using a semi-Lagrangian WENO scheme implemented on GPU

A numerical method for solving the Vlasov–Poisson equations using a high-order semi-Lagrange conservative WENO scheme is developed. The Vlasov–Poisson equations govern evolution of the collisionless self-interacting medium and are widely used in plasma physics and astrophysics, in particular for modeling dynamics of galactic systems. The method is implemented for computations on Graphical Processing Units (GPUs). The GPU code is validated using an exact unsteady analytical solution describing nonlinear oscillations of a plane self-gravitating layer. The comparison with numerical results obtained with the serial CPU code show a significant, up to 50 times, speed-up of the computations.

A general framework for the evaluation of shock-capturing schemes

We introduce a standardized procedure for benchmarking shock-capturing schemes which is intended to go beyond traditional case-by-case analysis, by setting objective metrics for cross-comparison of flow solvers. The main idea is that use of shock-capturing schemes yields both distributed errors associated with propagation of wave-like disturbances in smooth flow regions, and localized errors at shocks where nonlinear numerical mechanisms are most active. Our standardized error evaluation framework relies on previous methods of analysis for the propagation error with associated cost/error mapping, and on novel analysis of the shock-capturing error based on a model scalar problem. Amplitude and phase errors are identified for a number of classical shock-capturing schemes with different order of accuracy. Whereas all schemes are found to be, as expected, first-order accurate at shocks, quantitative differences are found to be significant, and we find that certain schemes in wide use (e.g. high-order WENO schemes) may yield substantial over-amplification of incoming disturbances at shocks. Illustrative calculations are also shown for the 1D Euler equations, which support sufficient generality of the analysis, although nonlinearity suggests caution in straightforward extrapolation to other flow cases.

Optimizing computational high-order schemes in finite volume simulations using unstructured mesh and topological data structures

Many numerical methods are based in mesh files to represent the computational domain. Also, an efficient storage and retrieval of mesh information can be achieved by data structures. Moreover, the development of topological operators is one of the important goals of the geometric modeling research field. While it loads mesh files more efficiently, it also allocates less main memory, provides persistence and allows consistent query operations. This paper proposes an improving of the computational scheme of high-order WENO schemes by coupling a standard cell centered, unstructured mesh, finite volume method with an topological data structure. The solver module uses the finite volume technique with a formulation that sets the property values to the control volume centroids. The two dimensional Euler equations are considered to represent the flow of interest. Beyond experiments using the improved approach, the computational cost of the data structure was measured by comparing with the traditional representation, and the results showed that our approach provides scalable loading and managing of meshes, having less memory occupation rate when comparing meshes with an increasing number of elements.

Simulation of a 3D Turbulent Wavy Channel based on the High-order WENO Scheme

Passive interest turbulent drag reduction, effective means to improve air vehicle fuel consumption costs. Most turbulent problems happening to the nature and engineering applications were exactly the turbulence problem frequently caused by one or more turbulent shear flows. This study was operated with incompressible 3-D channels with cyclic wavy boundary to explore the physical properties of turbulence flow. This research measures the distribution of average velocity, instant flowing field shapes, turbulence and pressure distribution, etc. Furthermore, the systematic computation and analysis for the 3-D flow field was also implemented. It was aimed to clearly understand the turbulence fields formed by wavy boundary of tube flow. The purpose of this research is to obtain systematic structural information about the turbulent flow field and features of the turbulence structure are discussed.

Efficient simulation of chromatographic separation processes

This work presents the development and testing of an efficient, high resolution algorithm developed for the solution of equilibrium and non-equilibrium chromatographic problems as a means of simultaneously producing high fidelity predictions with a minimal increase in computational cost. The method involves the coupling of a high-order WENO scheme, adapted for use on non-uniform grids, with a piecewise adaptive grid (PAG) method to reduce runtime while accurately resolving the sharp gradients observed in the processes under investigation. Application of the method to a series of benchmark chromatographic test cases, within which an increasing number of components are included over short and long spatial domains and containing shocks, shows that the method is able to accurately resolve the discontinuities and that the use of the PAG method results in a reduction in the CPU runtime of up to 90%, without degradation of the solution, relative to an equivalent uniform grid.

Accelerating high-order WENO schemes using two heterogeneous GPUs

A double-GPU code is developed to accelerate WENO schemes. The test problem is a compressible viscous flow. The convective terms are discretized using third- to ninth-order WENO schemes and the viscous terms are discretized by the standard fourth-order central scheme. The code written in CUDA programming language is developed by modifying a single-GPU code. The OpenMP library is used for parallel execution of the code on both the GPUs. Data transfer between GPUs which is the main issue in developing the code, is carried out by defining halo points for numerical grids and by using a CUDA built-in function. The code is executed on a PC equipped with two heterogeneous GPUs. The computational times of different schemes are obtained and the speedups with respect to the single-GPU code are reported for different number of grid points. Furthermore, the developed code is analyzed by CUDA profiling tools. The analyze helps to further increase the code performance.

Fine structures for the solutions of the two-dimensional Riemann problems by high-order WENO schemes

The two-dimensional Riemann problem with polytropic gas is considered. By a restriction on the constant states of each quadrant of the computational domain such that there is only one planar centered wave connecting two adjacent quadrants, there are nineteen genuinely different initial configurations of the problem. The configurations are numerically simulated on a fine grid and compared by the 5th-order WENO-Z5, 6th-order WENO-źź6, and 7th-order WENO-Z7 schemes. The solutions are very well approximated with high resolution of waves interactions phenomena and different types of Mach shock reflections. Kelvin-Helmholtz instability-like secondary-scaled vortices along contact continuities are well resolved and visualized. Numerical solutions show that WENO-źź6 outperforms the comparing WENO-Z5 and WENO-Z7 in terms of shock capturing and small-scaled vortices resolution. A catalog of the numerical solutions of all nineteen configurations obtained from the WENO-źź6 scheme is listed. Thanks to their excellent resolution and sharp shock capturing, the numerical solutions presented in this work can be served as reference solutions for both future numerical and theoretical analyses of the 2D Riemann problem.

Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton-Jacobi equations

In this paper, a class of high-order central Hermite WENO (HWENO) schemes based on finite volume framework and staggered meshes is proposed for directly solving one- and two-dimensional Hamilton-Jacobi (HJ) equations. The methods involve the Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time. This work can be regarded as an extension of central HWENO schemes for hyperbolic conservation laws (Tao et al. J. Comput. Phys. 318, 222–251, 2016) which combine the central scheme and the HWENO spatial reconstructions and therefore carry many features of both schemes. Generally, it is not straightforward to design a finite volume scheme to directly solve HJ equations and a key ingredient for directly solving such equations is the reconstruction of numerical Hamiltonians to guarantee the stability of methods. Benefited from the central strategy, our methods require no numerical Hamiltonians. Meanwhile, the zeroth-order and the first-order moments of the solution are involved in the spatial HWENO reconstructions which is more compact compared with WENO schemes. The reconstructions are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. A collection of one- and two- dimensional numerical examples is performed to validate high resolution and robustness of the methods in approximating the solutions of HJ equations, which involve linear, nonlinear, smooth, non-smooth, convex or non-convex Hamiltonians.

Detonation wave diffraction in H2–O2–Ar mixtures

In the present study, we have examined the diffraction of detonation in weakly unstable hydrogen–oxygen–argon mixtures. High accuracy and computational efficiency are obtained using a high-order WENO scheme together with adaptive mesh refinement, which enables handling realistic geometries with resolution at the micrometer level. Both detailed chemistry and spectroscopic models of laser induced fluorescence and chemiluminescence were included to enable a direct comparison with experimental data. Agreement was found between the experiments and the simulations in terms of detonation diffraction structure both for sub-critical and super-critical regimes. The predicted wall reflection distance is about 12–14 cell widths, in accordance with previous experimental studies. Computations show that the re-initiation distance is relatively constant, at about 12–15 cell widths, slightly above the experimental value of 11 cell widths. The predicted critical channel height is 10–11 cell widths, which differs from experiments in circular tubes but is consistent with rectangular channel results.

Parametric Investigation of Breaking Solitary Wave Over Fringing Reef Based on Shock-Capturing Boussinesq Model

In this paper, a hybrid finite-volume-finite-difference scheme for the solution of the one-dimensional form of the Nwogu's extended Boussinesq-type equations was developed to investigate the breaking of a solitary wave propagating over a fringing reef system. The adoption of a high-order finite volume WENO schemes with HLL Riemann solver for advection terms as well as the third-order Runge—Kutta time-stepping scheme makes the present scheme shock-capturing. From the validation case, it is shown that the present shock-capturing model could simulate the shoreline movement and wave breaking as well as bore propagation very well. In order to comprehensively understand the hydrodynamic characteristics when a solitary wave travels over a fringing reef system, a parametric study has been carried out to analyze the interaction process under various conditions. Effects of different physical parameters on wave breaking axe analyzed and the reflection, transmission and dissipation (RTD) coefficients are calculated based on the integration of kinetic and potential energies in different zones.

High-order finite difference WENO schemes with positivity-preserving limiter for correlated random walk with density-dependent turning rates

In this paper, we discuss high-order finite difference weighted essentially non-oscillatory schemes, coupled with total variation diminishing (TVD) Runge–Kutta (RK) temporal integration, for solving the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. Since the solutions to this system are non-negative, we discuss a positivity-preserving limiter without compromising accuracy. Analysis is performed to justify the maintenance of third-order spatial/temporal accuracy when the limiters are applied to a third-order finite difference scheme and third-order TVD-RK time discretization for solving this model. Numerical results are also provided to demonstrate these methods up to fifth-order accuracy.

Smooth Particle Hydrodynamics with nonlinear Moving-Least-Squares WENO reconstruction to model anisotropic dispersion in porous media

Most numerical schemes applied to solve the advection–diffusion equation are affected by numerical diffusion. Moreover, unphysical results, such as oscillations and negative concentrations, may emerge when an anisotropic dispersion tensor is used, which induces even more severe errors in the solution of multispecies reactive transport. To cope with this long standing problem we propose a modified version of the standard Smoothed Particle Hydrodynamics (SPH) method based on a Moving-Least-Squares-Weighted-Essentially-Non-Oscillatory (MLS-WENO) reconstruction of concentrations. This scheme formulation (called MWSPH) approximates the diffusive fluxes with a Rusanov-type Riemann solver based on high order WENO scheme. We compare the standard SPH with the MWSPH for different a few test cases, considering both homogeneous and heterogeneous flow fields and different anisotropic ratios of the dispersion tensor. We show that, MWSPH is stable and accurate and that it reduces the occurrence of negative concentrations compared to standard SPH. When negative concentrations are observed, their absolute values are several orders of magnitude smaller compared to standard SPH. In addition, MWSPH limits spurious oscillations in the numerical solution more effectively than classical SPH. Convergence analysis shows that MWSPH is computationally more demanding than SPH, but with the payoff a more accurate solution, which in addition is less sensitive to particles position. The latter property simplifies the time consuming and often user dependent procedure to define the initial dislocation of the particles.

A Maximum-Principle-Satisfying High-Order Finite Volume Compact WENO Scheme for Scalar Conservation Laws with Applications in Incompressible Flows

In this paper, a maximum-principle-satisfying finite volume compact scheme is proposed for solving scalar hyperbolic conservation laws. The scheme combines weighted essentially non-oscillatory schemes (WENO) with a class of compact schemes under a finite volume framework, in which the nonlinear WENO weights are coupled with lower order compact stencils. The maximum-principle-satisfying polynomial rescaling limiter in Zhang and Shu (J Comput Phys 229:3091---3120, 2010, Proc R Soc A Math Phys Eng Sci 467:2752---2776, 2011) is adopted to construct the present schemes at each stage of an explicit Runge---Kutta method, without destroying high order accuracy and conservativity. Numerical examples for one and two dimensional problems including incompressible flows are presented to assess the good performance, maximum principle preserving, essentially non-oscillatory and high resolution of the proposed method.

High-order central Hermite WENO schemes on staggered meshes for hyperbolic conservation laws

In this paper, we propose a class of high-order schemes for solving one- and two-dimensional hyperbolic conservation laws. The methods are formulated in a central finite volume framework on staggered meshes, and they involve Hermite WENO (HWENO) reconstructions in space, and Lax–Wendroff type discretizations or the natural continuous extension of Runge–Kutta methods in time. Compared with central WENO methods, the spatial reconstruction used here is much more compact; and unlike the original HWENO methods, our proposed schemes require neither flux splitting nor the use of numerical fluxes. In the system case, local characteristic decomposition is applied in the reconstructions of cell averages to enhance the non-oscillatory property of the methods. The high resolution and robustness of the methods in capturing smooth and non-smooth solutions are demonstrated through a collection of one- and two-dimensional scalar and system of examples.

High-order well-balanced central WENO scheme for pre-balanced shallow water equations

In this paper, we are concerned with numerically solving shallow water equations with a source term. The equations admit steady state solutions in which the non-zero flux gradient is exactly balanced by the source term. It is difficult for numerical schemes to preserve the delicate balance between the flux gradient and the source term.By virtue of a modified governing equations as well as an original reconstruction procedure for the Runge–Kutta flux, we present a central weighted essentially non-oscillatory scheme. Extensive test cases are carried out to demonstrate the performance of the resulting scheme: exact conservation property, high-order accuracy and good resolution near discontinuities.

On one-dimensional arbitrary high-order WENO schemes for systems of hyperbolic conservation laws

In this paper, a one-dimensional case for an arbitrary high-order non-oscillatory finite-volume scheme is considered. This is an adaptation of the schemes presented in Dumbser and Kaser (J Comput Phys. 221:693–723, 2007). An entire polynomial inside each control volume is reconstructed in terms of hierarchical orthogonal polynomial bases over a reference element. A new numerical inter-cell flux function at the element interfaces is proposed. To validate the approach, several one-dimensional test problems with discontinuous solutions are computed. The results show an improvement on the non-oscillatory shock-capturing properties based on the new approach.

Viscous effects on the non-classical Rayleigh–Taylor instability of spherical material interfaces

The viscous and conductivity effects on the instability of a rapidly expanding material interface produced by a spherical shock tube are investigated through the employment of a high-order WENO scheme. The instability is influenced by various mechanisms, which include (a) classical Rayleigh–Taylor (RT) effects, (b) Bell–Plesset or geometry/curvature effects, (c) the effects of impulsively accelerating the interface, (d) compressibility effects, (e) finite thickness effects, and (f) viscous effects. Henceforth, the present instability studied is more appropriately referred to as non-classical RT instability to distinguish it from classical RT instability. The linear regime is examined and the development of the viscous three-dimensional perturbations is obtained by solving a one-dimensional system of partial differential equations. Numerical simulations are performed to illustrate the viscous effects on the growth of the disturbances for various conditions. The inviscid analysis does not show the existence of a maximum amplification rate. The present viscous analysis, however, shows that the growth rate increases with increasing the wave number, but there exists a peak wavenumber beyond which the growth rate decreases with increasing the wave number due to viscous effects.

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

An E-CUSP (energy-convective upwind and split pressure) scheme is developed to solve the equations of magnetohydrodynamics. Fifth order WENO reconstructions are employed to calculate the fluxes in order to achieve high order spacial accuracy. A characteristic speed of sound by averaging the fast wave speed and the acoustic speed of sound is suggested to evaluate the Mach number, which will yield robust and accurate solutions. The numerical experiments have demonstrated the accuracy and the capability of the new scheme to capture complex interactions of multiple shocks and vortices.

Representation of the Lagrange reconstructing polynomial by combination of substencils

The Lagrange reconstructing polynomial [C.W. Shu, High-order WENO schemes for convection-dominated problems, SIAM Rev. 51 (1) (2009) 82–126] of a function f(x) on a given set of equidistant (Δx=const) points {xi+ℓΔx;ℓ∈{−M−,…,+M+}} is defined as the polynomial whose sliding (with x) averages on [x−12Δx,x+12Δx] are equal to the Lagrange interpolating polynomial of f(x) on the same stencil [G.A. Gerolymos, Approximation error of the Lagrange reconstructing polynomial, J. Approx. Theory 163 (2) (2011) 267–305. doi:10.1016/j.jat.2010.09.007]. We first study the fundamental functions of Lagrange reconstruction, then show that these polynomials have only real and distinct roots, which are never located at the cell-interfaces (half-points) xi+n12Δx (n∈Z), and obtain several identities. Using these identities, we show that there exists a unique representation of the Lagrange reconstructing polynomial on {i−M−,…,i+M+} as a combination of the Lagrange reconstructing polynomials on Neville substencils [E. Carlini, R. Ferretti, G. Russo, A WENO large time-step scheme for Hamilton–Jacobi equations, SIAM J. Sci. Comput. 27 (3) (2005) 1071–1091], with weights which are rational functions of ξ (x=xi+ξΔx) [Y.Y. Liu, C.W. Shu, M.P. Zhang, On the positivity of the linear weights in WENO approximations, Acta Math. Appl. Sin. 25 (3) (2009) 503–538], and give an analytical recursive expression of the weight-functions. We show that all of the poles of the rational weight-functions are real, and that there can be no poles at half-points. We then use the analytical expression of the weight-functions, combined with the factorization of the fundamental functions of Lagrange reconstruction, to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of ξ=12, iff all of the substencils contain either point i or point i+1 (or both).