Weighted Essentially Non-oscillatory Interpolation Research Articles

High order conservative finite difference WENO scheme for three-temperature radiation hydrodynamics

The three-temperature (3-T) radiation hydrodynamics (RH) equations play an important role in the high-energy-density-physics fields, such as astrophysics and inertial confinement fusion (ICF). It describes the interaction between radiation and high-energy-density plasmas including electron and ion in the assumption that radiation, electron and ion are in their own equilibrium state, which means they can be characterized by their own temperatures. The 3-T RH system consists of the density, momentum and three internal energy (electron, ion and radiation) equations. In this paper, we propose a high order conservative finite difference weighted essentially non-oscillatory (WENO) scheme solving one-dimensional (1D) and two-dimensional (2D) 3-T RH equations respectively. Following our previous paper [7], we introduce the three new energy variables, and then design a finite difference scheme with both the conservative property and arbitrary high order accuracy. Based on the WENO interpolation and the strong stability preserving (SSP) high order time discretizations, taken as an example, we design a class of fifth order conservative finite difference schemes in space and third order in time. Compared with the Lagrangian method we proposed in [7], which can only reach second order accuracy for 2D 3-T RH equations if straight-line edged meshes are used, the finite difference scheme can be easily designed to arbitrary high order accuracy for multi-dimensional 3-T RH equations. The finite difference formulation is also much less expensive in multi-dimensions than finite volume schemes used in [7]. Furthermore, our method can handle fluids with large deformation easily. Numerous 1D and 2D numerical examples are presented to verify the desired properties of the high order finite difference WENO schemes such as high order accuracy, non-oscillation, conservation and adaptation to severely distorted single-material radiation hydrodynamics problems.

Sixth-order perturbed WENO interpolation-based AWENO and WCNS-E schemes for hyperbolic conservation laws

The weighted essentially non-oscillatory (WENO) interpolation-based schemes (the alternative WENO (AWENO) scheme and the explicit weighted compact nonlinear (WCNS-E) scheme) are limited to the fifth-order despite the sixth-order Taylor expansion of the numerical flux used. This order reduction is due to the fifth-order accuracy inherent in the WENO interpolation. We investigate the perturbed WENO interpolation with a free parameter and affine-invariant WENO weights to recover sixth-order accuracy in smooth regions. A cutoff function of the free parameter with a threshold, determined by approximate dispersion relation analysis, is applied to enhance the ENO property. The proposed schemes perform better in accuracy, dissipation, resolution, shock-capturing, and efficiency in 1D and 2D benchmark problems.

Constrained least-squares based adaptive-order finite-volume WENO scheme for the simulation of viscous compressible flows on unstructured grids

We develop a novel constrained least-squares technique for fourth-order weighted essentially non-oscillatory (WENO) polynomial reconstruction on general unstructured grids composed of curved-edged quadrilateral elements. Our technique relies on a segregation of the conditions that match the spatial averages of the candidate WENO polynomial and the reconstructed state variable over stencil-forming cells into two sets. The first set consists of exactly enforced linearly independent constraints that are applied only over the central cell and its edge-sharing neighbors. The second set consists of the remaining conditions and are enforced only approximately in a least-squares sense. We develop a specialized technique that reduces the computational burden associated with the numerical solution of the constrained least-squares system for the WENO interpolation coefficients. For a comparable computational expense, our constrained reconstruction approach yields pointwise estimates that are consistently more accurate than the conventional least-squares based estimates over a wide spectrum of spatial wavenumbers. This enhanced spectral resolution yields a significant, over two-fold increase in the accuracy with which unsteady smooth flow features, such as an advecting isentropic vortex, are captured over highly distorted curvilinear unstructured grids. Our constraint based reconstruction framework is generic, facilitates implementation of high-order boundary closures, and is amenable to large scale parallelization over several thousand CPU cores. Numerical tests over a range of inviscid and viscous flow configurations are presented to demonstrate the accuracy and robustness of our technique. Our constrained least-squares based adaptive WENO discretization faithfully captures intricate features that arise in strongly shocked, and highly unsteady and separated complex geometry flows. Our results suggest exact preservation of the match between the cell averages of the interpolant and the reconstructed state variable over the nearest neighbors to be quite an effective strategy that dramatically enhances the accuracy and spectral resolution of the high-order finite-volume reconstruction operation.

Fast Sparse Grid Simulations of Fifth Order WENO Scheme for High Dimensional Hyperbolic PDEs

The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order WENO schemes, are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers’ equations. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. Several open problems are discussed at last.

WENO interpolation for Lagrangian particles in highly compressible flow regimes

Lagrangian particles can be used either to analyze different complex flows, as massless tracers, or to model multiphase flows, as particles with mass. As particles can have arbitrary positions within the computational domain, interpolation of different flow quantities from the Eulerian grid is essential. Low-order centered interpolation schemes generally do not provide a sufficient level of accuracy for many flow configurations of interest. Thus, typically, high-order centered interpolation schemes are utilized. The current study demonstrates that for highly compressible non-reacting and reacting flow regimes, in which discontinuities in the flow field, e.g., shock waves arise, centered interpolation schemes tend to smear the shock and high order schemes also produce numerical oscillations. It is shown that this problem can be remedied by using Weighted-Essentially-Non-Oscillatory (WENO) interpolation schemes. Extensive numerical tests are performed in order to demonstrate this, comparing the performance of WENO-3 and WENO-5 interpolation schemes with a variety of centered interpolation schemes. The first test case involves specially designed steady state two-dimensional vortex flows. For a smooth flow, WENO schemes provide comparable accuracy to other high-order centered schemes. For flow fields with discontinuities, WENO-3 and WENO-5 interpolation schemes can decrease the interpolation error by more than two and three orders of magnitude, respectively, in comparison with centered schemes. Solution accuracy is further studied with a normal shock wave test. It is demonstrated that centered schemes tend to smear the shock, whereas, WENO schemes capture it in a much sharper manner. Moreover, high-order centered schemes tend to oscillate in the vicinity of the discontinuity leading to non-physical results, such as negative absolute pressure values. The same trends are retained for an unsteady three-dimensional spherical blast wave, where the shock is smeared across several grid cells, which is typical for shock-capturing flow solvers. Finally, the benefits of WENO schemes for the Lagrangian-particle tracking analysis of highly compressible reactive flows are explored by comparing various Lagrangian particle trajectories and joint probability density functions (PDFs) for a two-dimensional cellular detonation. The results indicate that high-order centered schemes lead to unphysical results, whereas WENO schemes can provide high-order interpolation that is free of severe numerical oscillations and non-physical artifacts, which is critical for the proper analysis of the flow.

A novel fourth-order WENO interpolation technique

Context. Several numerical problems require the interpolation of discrete data that present at the same time (i) complex smooth structures and (ii) various types of discontinuities. The radiative transfer in solar and stellar atmospheres is a typical example of such a problem. This calls for high-order well-behaved techniques that are able to interpolate both smooth and discontinuous data. Aims. This article expands on different nonlinear interpolation techniques capable of guaranteeing high-order accuracy and handling discontinuities in an accurate and non-oscillatory fashion. The final aim is to propose new techniques which could be suitable for applications in the context of numerical radiative transfer. Methods. We have proposed and tested two different techniques. Essentially non-oscillatory (ENO) techniques generate several candidate interpolations based on different substencils. The smoothest candidate interpolation is determined from a measure for the local smoothness, thereby enabling the essentially non-oscillatory property. Weighted ENO (WENO) techniques use a convex combination of all candidate substencils to obtain high-order accuracy in smooth regions while keeping the essentially non-oscillatory property. In particular, we have outlined and tested a novel well-performing fourth-order WENO interpolation technique for both uniform and nonuniform grids. Results. Numerical tests prove that the fourth-order WENO interpolation guarantees fourth-order accuracy in smooth regions of the interpolated functions. In the presence of discontinuities, the fourth-order WENO interpolation enables the non-oscillatory property, avoiding oscillations. Unlike Bézier and monotonic high-order Hermite interpolations, it does not degenerate to a linear interpolation near smooth extrema of the interpolated function. Conclusion. The novel fourth-order WENO interpolation guarantees high accuracy in smooth regions, while effectively handling discontinuities. This interpolation technique might be particularly suitable for several problems, including a number of radiative transfer applications such as multidimensional problems, multigrid methods, and formal solutions.

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.

Assessment of WENO-extended two-fluid modelling in compressible multiphase flows

ABSTRACTThe two-fluid modelling based on an advection-upwind-splitting-method (AUSM)-family numerical flux function, AUSM+-up, following the work by Chang and Liou [Journal of Computational Physics 2007;225: 840–873], has been successfully extended to the fifth order by weighted-essentially-non-oscillatory (WENO) schemes. Then its performance is surveyed in several numerical tests. The results showed a desired performance in one-dimensional benchmark test problems: Without relying upon an anti-diffusion device, the higher-order two-fluid method captures the phase interface within a fewer grid points than the conventional second-order method, as well as a rarefaction wave and a very weak shock. At a high pressure ratio (e.g. 1,000), the interpolated variables appeared to affect the performance: the conservative-variable-based characteristic-wise WENO interpolation showed less sharper but more robust representations of the shocks and expansions than the primitive-variable-based counterpart did. In two-dimensional shock/droplet test case, however, only the primitive-variable-based WENO with a huge void fraction realised a stable computation.

An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers’ equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme.

Unified Gas Kinetic Scheme and Direct Simulation Monte Carlo Computations of High-Speed Lid-Driven Microcavity Flows

AbstractAccurate simulations of high-speed rarefied flows present many physical and computational challenges. Toward this end, the present work extends the Unified Gas Kinetic Scheme (UGKS) to a wider range of Mach and Knudsen numbers by implementing WENO (Weighted Essentially Non-Oscillatory) interpolation. Then the UGKS is employed to simulate the canonical problem of lid-driven cavity flow at high speeds. Direct Simulation Monte Carlo (DSMC) computations are also performed when appropriate for comparison. The effect of aspect ratio, Knudsen number and Mach number on cavity flow physics is examined leading to important insight.

An Alternative Formulation of Finite Difference Weighted ENO Schemes with Lax--Wendroff Time Discretization for Conservation Laws

We develop an alternative formulation of conservative finite difference weighted essentially nonoscillatory (WENO) schemes to solve conservation laws. In this formulation, the WENO interpolation of the solution and its derivatives are used to directly construct the numerical flux, instead of the usual practice of reconstructing the flux functions. Even though this formulation is more expensive than the standard formulation, it does have several advantages. The first advantage is that arbitrary monotone fluxes can be used in this framework, while the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. The second advantage, which is fully explored in this paper, is that a narrower effective stencil is used compared with previous high order finite difference WENO schemes based on the reconstruction of flux functions, with a Lax--Wendroff time discretization. We will describe the scheme formulation and present numerical tests for one- and two-dimensional scalar ...

Non-separable two-dimensional weighted ENO interpolation

The one-dimensional weighted essentially non-oscillatory (WENO) point-value interpolation proposed in Aràndiga et al. (2010) [4] generically achieves an order of accuracy of 2r, when using stencils of 2r points at regions where the interpolated function is smooth. In this paper we propose a non-separable two-dimensional weighted ENO interpolation that extends in a non-tensorial manner the above mentioned WENO interpolation and has the same order of accuracy at smooth regions.The image zoom capability of this interpolation is compared to those of linear and WENO tensorial interpolation.

On the efficient application of weighted essentially nonoscillatory scheme

SUMMARYIn this paper, the efficient application of high‐order weighted essentially nonoscillatory (WENO) reconstruction to the subsonic and transonic engineering problems is studied. On the basis of the physical considerations, two techniques are proposed to enhance the accuracy and efficiency of the WENO reconstruction. First, it is observed that the WENO scheme using characteristic variable has better accuracy and convergence speed than the scheme using primitive variable. For engineering problems with shock of moderate amplitude, on the basis of the Rankine–Hugoniot conditions, a simplified characteristic‐variable‐based WENO is developed. The simplified version significantly reduces the cost overhead without sacrificing the shock‐capturing capability. Second, in this work, it is found for viscous case that it is better to include the viscous effect. On the basis of a simple analysis, the viscous correction to the parameter ε in the WENO reconstruction is proposed. Numerical results indicate, with the proposed simplified characteristic‐variable‐based reconstruction and the viscous correction, that the nonlinear WENO interpolation is sharply activated in the region of shock jump, whereas in the shockless area, the WENO interpolation weights are tuned towards the designed optimal value for better accuracy. Compared with the original characteristic‐variable‐based WENO, the current implementation has similar accuracy and reduced cost. At the same time, compared with the primitive variable‐based WENO, better accuracy and convergence speed are obtained at marginal cost overhead. Several practical cases are calculated to demonstrate the accuracy and efficiency of the current methodology. Copyright © 2012 John Wiley & Sons, Ltd.

Binary weighted essentially non-oscillatory (BWENO) approximation

Weighted essentially non-oscillatory (WENO) schemes have been mainly used for solving hyperbolic partial differential equations (PDEs). Such schemes are capable of high order approximation in smooth regions and non-oscillatory sharp resolution of discontinuities. The base of the WENO schemes is a non-oscillatory WENO approximation procedure, which is not necessarily related to PDEs. The typical WENO procedures are WENO interpolation and WENO reconstruction. The WENO algorithm has gained much popularity but the basic idea of approximation did not change much over the years. In this paper, we first briefly review the idea of WENO interpolation and propose a modification of the basic algorithm. New approximation should improve basic characteristics of the approximation and provide a more flexible framework for future applications. New WENO procedure involves a binary tree weighted construction that is based on key ideas of WENO algorithm and we refer to it as the binary weighted essentially non-oscillatory (BWENO) approximation. New algorithm comes in a rational and a polynomial version. Furthermore, we describe the WENO reconstruction procedure, which is usually involved in the numerical schemes for hyperbolic PDEs, and propose the new reconstruction procedure based on the described BWENO interpolation. The obtained numerical results show that the newly proposed procedures perform very well on the considered test examples.

Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations

In this paper, we propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework [4,5] with the high order finite difference weighted essentially non-oscillatory (WENO) method in space and the total variation diminishing (TVD) Runge–Kutta (RK) method in time (WENO-RK) [18,10] by a high order coupling. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. The high order coupling of AMR and WENO-RK is accomplished by high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from coarse to fine grid solutions, and at ghost points. The resulting AMR-WENO method is accurate, robust and efficient, due to the mesh adaptivity and very high order spatial and temporal accuracy. We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes. In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.

On the positivity of linear weights in WENO approximations

High order accurate weighted essentially non-oscillatory (WENO) schemes have been used extensively in numerical solutions of hyperbolic partial differential equations and other convection dominated problems. However the WENO procedure can not be applied directly to obtain a stable scheme when negative linear weights are present. In this paper, we first briefly review the WENO framework and the role of linear weights, and then present a detailed study on the positivity of linear weights in a few typical WENO procedures, including WENO interpolation, WENO reconstruction and WENO approximation to first and second derivatives, and WENO integration. Explicit formulae for the linear weights are also given for these WENO procedures. The results of this paper should be useful for future design of WENO schemes involving interpolation, reconstruction, approximation to first and second derivatives, and integration procedures.

Weighted ENO interpolation and applications

Data-dependent interpolatory techniques such as essentially non-oscillatory (ENO) technique [J. Comput. Phys. 71 (1987) 231] have long been used as a reconstruction process in multiresolution schemes. In this work we analyze the weighted ENO (WENO) technique introduced by Liu et al. in the context of conservation laws [J. Comput. Phys. 115 (1994) 200] and improved by Jiang and Shu [J. Comput. Phys. 126 (1996) 202], and apply it to the compression of images, using multiresolution techniques.

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.