Accuracy In Smooth Regions Research Articles

A high performance fifth‐order multistep WENO scheme

SummaryMany efforts have been made to improve the accuracy of the conventional weighted essentially nonoscillatory (WENO) scheme at transition points (connecting a smooth region and a discontinuity point). This paper analyzes these works and further develops a more effective multistep WENO scheme. Theoretical analysis and numerical results show that the new scheme not only improves the accuracy by one order higher than the traditional fifth‐order WENO schemes at transition point but also maintains the fifth‐order accuracy in smooth regions even at critical point where the first derivative vanishes.

A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes

In this paper, we continue our work in [46] and propose a new type of high-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. Although termed “multi-resolution WENO schemes”, we only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. We construct new third-order, fourth-order, and fifth-order WENO schemes using three or four unequal-sized central spatial stencils, different from the classical WENO procedure using equal-sized biased/central spatial stencils for the spatial reconstruction. The new WENO schemes could obtain the optimal order of accuracy in smooth regions, and could degrade gradually to first-order of accuracy so as to suppress spurious oscillations near strong discontinuities. This is the first time that only a series of unequal-sized hierarchical central spatial stencils are used in designing arbitrary high-order finite volume WENO schemes on triangular meshes. The main advantages of these schemes are their compactness, robustness, and their ability to maintain good convergence property for steady-state computation. The linear weights of such WENO schemes can be any positive numbers on the condition that they sum to one. Extensive numerical results are provided to illustrate the good performance of these new finite volume WENO schemes.

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.

Adaptive mapping for high order WENO methods

In this paper, a novel mapping approach through the use of adaptive mapping functions is introduced for high order weighted essentially non-oscillatory (WENO) methods. The new class of adaptive mapping functions are designed to adjust themselves to the solution based on a simple parameter calculated using the smoothness indicators that are readily available during computation. It is shown that this adaptive nature allows the resultant mapped WENO scheme to maintain sub-stencil weights close to the optimal weights in smooth regions without amplifying the weights of non-smooth stencils containing discontinuities. Therefore, adaptive mapping achieves enhanced accuracy in smooth regions and is more resistant against spurious oscillations near discontinuities. Taylor series analysis of the seventh order finite volume WENO scheme has been performed to demonstrate the loss of accuracy of the original WENO method near critical points. The convergence rates of the seventh order finite volume WENO scheme with adaptive mapping have been shown through a simple numerical example. Excellent results have been obtained for one-dimensional linear advection cases especially over long output times. Improved results have also been obtained for one- and two-dimensional Euler equation test cases.

Development of a mass-preserving level set redistancing algorithm for simulation of rising bubble

This article is aimed to simulate the gas-liquid flow of rising bubbles with a mass-preserving level set method. To resolve the topological changes of gas-liquid interface where the classic finite difference scheme may yield oscillation solutions, the spatial terms in the level set advection equation will be approximated by an optimized compact reconstruction weighted essentially non-oscillatory (OCRWENO) scheme. This scheme achieves high-order accuracy in smooth regions, and meanwhile avoid numerical oscillation near discontinuities. Two benchmark problems including vortex flow and deforming field are chosen to compare the present simulation with previous numerical researches. Several rising bubble problems are validated by the proposed level set method.

High order finite-volume WENO scheme for five-equation model of compressible two-fluid flow

High order finite-volume weighted essentially non-oscillatory (WENO) scheme is applied for solving interface-capturing five-equation model of compressible two-fluid flows in one and two-space dimensions. The model is non-conservative and the governing equations consist of three equations, namely a continuity equation, a momentum equation and an energy balance equation for the fluid mixture and the remaining two are mass and energy equations for one of the two fluids. In the last equation, the non-conservative differential source term appears which is responsible for the energy exchange between fluids. The energy exchange is only due to mechanical work. The presence of non-conservative differential source terms in the two-fluid flow model introduce difficulties in developing high order accurate numerical schemes. The proposed numerical scheme is capable to preserve non-oscillatory property near strong discontinuities and gives high order accuracy in smooth regions. Different one and two-dimensional test problems are considered to analyze efficiency and accuracy of the proposed numerical algorithms. For validation, the solutions of proposed numerical scheme are compared with the results of already available high order kinetic flux-vector splitting scheme and discontinuous Galerkin scheme.

A new type of multi-resolution WENO schemes with increasingly higher order of accuracy

In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in [25,45], could obtain the optimal order of accuracy in smooth regions, and could simultaneously suppress spurious oscillations near discontinuities. The linear weights of such WENO schemes can be any positive numbers on the condition that their sum equals one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference and finite volume WENO schemes. These new WENO schemes are simple to construct and can be easily implemented to arbitrary high order of accuracy and in higher dimensions. Benchmark examples are given to demonstrate the robustness and good performance of these new WENO schemes.

A non‐oscillatory multimoment finite‐volume global transport model on a cubed‐sphere grid using the WENO slope limiter

By using CSL3 multimoment interpolation, a piecewise cubic polynomial for spatial reconstruction can be obtained with four multimoment constraint conditions consisting of two point values at cell boundaries, one volume‐integrated average and one slope parameter at the cell center. The resulting multimoment finite‐volume scheme is of fourth‐order accuracy. A non‐oscillatory scheme can be derived by designing the proper formula to calculate the slope parameter at the cell center. A new strategy was recently proposed, using the Weighted Essentially Non‐Oscillatory (WENO) concept to determine the slope parameter. Using a WENO‐type limiter, the multimoment reconstruction can effectively remove nonphysical oscillations while keeping fourth‐order accuracy in smooth regions. In this study, a WENO‐type slope limiter is proposed and implemented in our multimoment finite‐volume global transport model based on the cubed‐sphere grid. The widely used benchmark tests, including both solid rotation and complicated deformational advection cases, are checked to verify the performance of the proposed global transport model. Numerical results reveal that a WENO‐type slope limiter can greatly improve the accuracy of the multimoment finite‐volume model compared with the former Total Variation Diminishing (TVD)‐type limiter. Furthermore, the proposed limiter is constructed over a compact stencil of only three adjacent cells. Without any user‐defined or problem‐dependent parameters, the present model is very promising for practical applications.

High order well-balanced discontinuous Galerkin methods for shallow water flow under temperature fields

In this paper, we present high order discontinuous Galerkin methods for the Ripa model. This model maintains the steady state. To achieve the well-balanced property, we first reformulate the governing equations and propose to construct a novel source term approximation. In addition, we apply a flux modification technique to construct the numerical fluxes. Rigorous numerical analysis as well as extensive one- and two-dimensional numerical results all verify that the proposed methods are well-balanced for the steady state, keep genuinely high order accuracy in smooth regions for smooth solutions, and enjoy good resolutions for discontinuous solutions.

Improved weighted ENO scheme based on parameters involved in nonlinear weights

Here, we have analyzed the weights of the fifth-order finite difference weighted essentially non-oscillatory WENO-P scheme developed by Kim et al. (J. Sci. Comput. 2016) to approximate the solutions of hyperbolic conservation laws. The main ingredient of WENO schemes is the construction of smoothness indicators, which resolves odd behavior of the scheme near discontinuities. In WENO-P, the smoothness indicators are constructed in L1− norm. It is observed that analytically as well as numerically, the WENO-P weights do not achieve required ENO order of accuracy near discontinuities. To recover the desired order of accuracy, we have imposed some constraints on the weight parameters to guarantee that the WENO-P scheme achieves the desired ENO order of accuracy near discontinuities and have the over all fifth-order accuracy in smooth regions of solutions with an arbitrary number of vanishing derivatives. Numerical results are presented with the new weights to verify the robustness and accuracy of the proposed scheme for one and two-dimensional system of Euler equations.

High order well-balanced discontinuous Galerkin methods for Euler equations at isentropic equilibrium state under gravitational fields

Euler equations under gravitational fields often appear in some interesting astrophysical and atmospheric applications. The Euler equations are coupled with gravitational source term due to the gravity and admit hydrostatic equilibrium state where the flux produced by the pressure gradient is exactly balanced by the gravitational source term. In this paper, we construct high order discontinuous Galerkin methods for the Euler equations under gravitational fields, which are well-balanced for the isentropic type hydrostatic equilibrium state. To maintain the well-balanced property, we first reformulate the governing equations in an equivalent form. Then we propose a novel source term approximation based on a splitting algorithm as well as well-balanced numerical fluxes. Rigorous theoretical analysis and extensive numerical examples all suggest that the proposed methods maintain the hydrostatic equilibrium state up to the machine precision. Moreover, one- and two-dimensional simulations are performed to test the ability of the current methods to capture small perturbation of such equilibrium state, and the genuine high order accuracy in smooth regions.

A new weighting method for improving the WENO‐Z scheme

SummaryThe calculation of the weight of each substencil is very important for a weighted essentially nonoscillatory (WENO) scheme to obtain high‐order accuracy in smooth regions and keep the essentially nonoscillatory property near discontinuities. The weighting function introduced in the WENO‐Z scheme provides a straightforward method to analyze the accuracy order in smooth regions. In this paper, we construct a new sixth‐order global smoothness indicator (GSI‐6) and a function about GSI‐6 and the local smoothness indicators (ISk) to calculate the weights. The analysis and numerical results show that, with the new weights, the scheme satisfies the sufficient condition for the fifth‐order convergence in smooth regions even at critical points. Meanwhile, it can also maintain low dissipation for discontinuous solutions due to relative large weights assigned to discontinuous substencils.

Evaluation of numerical schemes for capturing shock waves in modeling proppant transport in fractures

In petroleum engineering, the transport phenomenon of proppants in a fracture caused by hydraulic fracturing is captured by hyperbolic partial differential equations (PDEs). The solution of this kind of PDEs may encounter smooth transitions, or there can be large gradients of the field variables. The numerical challenge posed in a shock situation is that high-order finite difference schemes lead to significant oscillations in the vicinity of shocks despite that such schemes result in higher accuracy in smooth regions. On the other hand, first-order methods provide monotonic solution convergences near the shocks, while giving poorer accuracy in the smooth regions. Accurate numerical simulation of such systems is a challenging task using conventional numerical methods. In this paper, we investigate several shock-capturing schemes. The competency of each scheme was tested against one-dimensional benchmark problems as well as published numerical experiments. The numerical results have shown good performance of high-resolution finite volume methods in capturing shocks by resolving discontinuities while maintaining accuracy in the smooth regions. These methods along with Godunov splitting are applied to model proppant transport in fractures. It is concluded that the proposed scheme produces non-oscillatory and accurate results in obtaining a solution for proppant transport problems.

A Novel Multi-Dimensional Limiter for High-Order Finite Volume Methods on Unstructured Grids

AbstractThis paper proposes a novel distance derivative weighted ENO (DDWENO) limiter based on fixed reconstruction stencil and applies it to the second- and highorder finite volume method on unstructured grids. We choose the standard deviation coefficients of the flow variables as the smooth indicators by using the k-exact reconstruction method, and obtain the limited derivatives of the flow variables by weighting all derivatives of each cell according to smoothness. Furthermore, an additional weighting coefficient related to distance is introduced to emphasize the contribution of the central cell in smooth regions. The developed limiter, combining the advantages of the slope limiters and WENO-type limiters, can achieve the similar effect of WENO schemes in the fixed stencil with high computational efficiency. The numerical cases demonstrate that the DDWENO limiter can preserve the numerical accuracy in smooth regions, and capture the shock waves clearly and steeply as well.

Numerical study on the convergence to steady state solutions of a new class of high order WENO schemes

A new class of high order weighted essentially non-oscillatory (WENO) schemes (Zhu and Qiu, 2016, [50]) is applied to solve Euler equations with steady state solutions. It is known that the classical WENO schemes (Jiang and Shu, 1996, [23]) might suffer from slight post-shock oscillations. Even though such post-shock oscillations are small enough in magnitude and do not visually affect the essentially non-oscillatory property, they are truly responsible for the residue to hang at a truncation error level instead of converging to machine zero. With the application of this new class of WENO schemes, such slight post-shock oscillations are essentially removed and the residue can settle down to machine zero in steady state simulations. This new class of WENO schemes uses a convex combination of a quartic polynomial with two linear polynomials on unequal size spatial stencils in one dimension and is extended to two dimensions in a dimension-by-dimension fashion. By doing so, such WENO schemes use the same information as the classical WENO schemes in Jiang and Shu (1996) [23] and yield the same formal order of accuracy in smooth regions, yet they could converge to steady state solutions with very tiny residue close to machine zero for our extensive list of test problems including shocks, contact discontinuities, rarefaction waves or their interactions, and with these complex waves passing through the boundaries of the computational domain.

Further improvement of weighted compact nonlinear scheme using compact nonlinear interpolation

To further improve the resolution of weighted compact nonlinear schemes(WCNS), a new 7th-order compact nonlinear interpolation method is proposed on the same stencil as 5th-order CRWENO scheme. Proper nonlinear weights are developed based on the Y type nonlinear weights, the properties of which are further analyzed in this paper. It is found that the Y type nonlinear weights are equivalent to the original nonlinear weights with an adaptive ϵ, which has very small value near discontinuities and very large value in smooth regions. As a result, the new nonlinear scheme has good shock capturing ability and achieves optimal order of accuracy in smooth regions. A new characteristic projection method is put forward, which largely reduces the computation costs. Simulations of Lax problem, Osher-Shu problem, double Mach reflection problem, decay of homogenous turbulence, shock turbulence interaction and turbulent channel flow show that the new scheme captures shocks without obvious oscillations, has higher resolution than other schemes tested, has obvious advantage in distinguishing turbulence from shocks and has higher efficiency.

Gradient‐based nodal limiters for artificial diffusion operators in finite element schemes for transport equations

SummaryThis paper presents new linearity‐preserving nodal limiters for enforcing discrete maximum principles in continuous (linear or bilinear) finite element approximations to transport problems with steep fronts. In the process of algebraic flux correction, the oscillatory antidiffusive part of a high‐order base discretization is decomposed into a set of internodal fluxes and constrained to be local extremum dim inishing. The proposed nodal limiter functions are designed to be continuous and satisfy the principle of linearity preservation that implies the preservation of second‐order accuracy in smooth regions. The use of limited nodal gradients makes it possible to circumvent angle conditions and guarantee that the discrete maximum principle holds on arbitrary meshes. A numerical study is performed for linear convection and anisotropic diffusion problems on uniform and distorted meshes in two space dimensions. Copyright © 2017 John Wiley & Sons, Ltd.

A new fifth order finite difference WENO scheme for Hamilton‐Jacobi equations

AbstractIn this continuing paper of (Zhu and Qiu, J Comput Phys 318 (2016), 110–121), a new fifth order finite difference weighted essentially non‐oscillatory (WENO) scheme is designed to approximate the viscosity numerical solution of the Hamilton‐Jacobi equations. This new WENO scheme uses the same numbers of spatial nodes as the classical fifth order WENO scheme which is proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143), and could get less absolute truncation errors and obtain the same order of accuracy in smooth region simultaneously avoiding spurious oscillations nearby discontinuities. Such new WENO scheme is a convex combination of a fourth degree accurate polynomial and two linear polynomials in a WENO type fashion in the spatial reconstruction procedures. The linear weights of three polynomials are artificially set to be any random positive constants with a minor restriction and the new nonlinear weights are proposed for the sake of keeping the accuracy of the scheme in smooth region, avoiding spurious oscillations and keeping sharp discontinuous transitions in nonsmooth region simultaneously. The main advantages of such new WENO scheme comparing with the classical WENO scheme proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143) are its efficiency, robustness and easy implementation to higher dimensions. Extensive numerical tests are performed to illustrate the capability of the new fifth WENO scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1095–1113, 2017

SpECTRE: A task-based discontinuous Galerkin code for relativistic astrophysics

We introduce a new relativistic astrophysics code, SpECTRE, that combines a discontinuous Galerkin method with a task-based parallelism model. SpECTRE's goal is to achieve more accurate solutions for challenging relativistic astrophysics problems such as core-collapse supernovae and binary neutron star mergers. The robustness of the discontinuous Galerkin method allows for the use of high-resolution shock capturing methods in regions where (relativistic) shocks are found, while exploiting high-order accuracy in smooth regions. A task-based parallelism model allows efficient use of the largest supercomputers for problems with a heterogeneous workload over disparate spatial and temporal scales. We argue that the locality and algorithmic structure of discontinuous Galerkin methods will exhibit good scalability within a task-based parallelism framework. We demonstrate the code on a wide variety of challenging benchmark problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the code's scalability including its strong scaling on the NCSA Blue Waters supercomputer up to the machine's full capacity of 22,380 nodes using 671,400 threads.

Entropy convergence of new two-value scheme with slope relaxation for conservation laws

This paper establishes the entropy convergence of a new two-value high resolution finite volume scheme with slope relaxation for conservation laws. This scheme, motivated by the general method of high resolution schemes that have high-order accuracy in smooth regions of solutions and are free of oscillations near discontinuities, unifies and evolves slopes directly with a slope relaxation equation that governs the evolution of slopes in both smooth and discontinuous regions. Proper choices of slopes are realized adaptively via a relaxation parameter. The scheme is shown to be total-variation-bounded (TVB) stable and satisfies cell-entropy inequalities.