Existing mapped WENO schemes can hardly prevent spurious oscillations while preserving high resolutions at long output times. We reveal in this paper the essential reason for such phenomena. It is actually caused by that the mapping function in these schemes can not preserve the order of the nonlinear weights of the stencils. The nonlinear weights may be increased for non-smooth stencils and be decreased for smooth stencils. It is then indicated to require the set of mapping functions to be Order-Preserving in mapped WENO schemes. Therefore, we propose a new mapped WENO scheme with a set of mapping functions to be order-preserving which exhibits a remarkable advantage over the mapped WENO schemes in references. For long output time simulations, the new scheme has the capacity to attain high resolutions and avoid spurious oscillations near discontinuities meanwhile.
The decisive factor for the calculation accuracy of the mapped weighted essentially non-oscillatory scheme is the width of the center region of the mapping function. Through analysis of the classical mapped WENO schemes, the results show the width of the central range of the mapping function determined by the local operator in its denominator. Substituting the local operator in WENO-AIM with a symmetric one and an asymmetric function, we get two new adaptive mapped WENO schemes, WENO-AIMS and WENO-AIMA. Similarly, we improve WENO-RM260 and WENO-PM6 by using these local operators, and composed …
A serious and ubiquitous issue in existing mapped WENO schemes is that most of them can hardly preserve high resolutions and in the meantime prevent spurious oscillations on solving hyperbolic conservation laws with long output times. Our goal in this article is to address this widely concerned problem [3, 4, 15, 29, 16, 18]. In our previous work [18], the order-preserving (OP) criterion was originally introduced and carefully used to devise a new mapped WENO scheme that performs satisfactorily in long-run simulations, and hence it was indicated that the OP criterion plays a critical role in t…
We propose a new family of mapped WENO schemes by using several adaptive control functions and a smoothing approximation of the signum function. The proposed schemes introduce the adaptivity and admit an extensive permitted range of the parameters in the mapping functions. Consequently, they have the capacity to achieve optimal convergence rates, even near critical points. Particularly, one of these new schemes with fine-tuned parameters illustrates a significant advantage when solving problems with discontinuities. It produces numerical solutions with high resolution without generating spurio…
We present a novel mapping approach for WENO schemes through the use of an approximate constant mapping function which is constructed by employing an approximation of the classic signum function. The new approximate constant mapping function is designed to meet the overall criteria for a proper mapping function required in the design of the WENO-PM6 scheme. The WENO-PM6 scheme was proposed to overcome the potential loss of accuracy of the WENO-M scheme which was developed to recover the optimal convergence order of the WENO-JS scheme at critical points. Our new mapped WENO scheme, denoted as W…
In our previous studies [17, 18], the commonly reported issue that most of the existing mapped WENO schemes suffer from either losing high resolutions or generating spurious oscillations in long-run simulations of hyperbolic problems has been successfully addressed, by devising the improved mapped WENO schemes, namely MOP-WENO-X, where "X" stands for the version of the existing mapped WENO scheme. However, all the MOP-WENO-X schemes bring about the serious deficiency that their resolutions in the region with high-frequency but smooth waves are dramatically decreased compared to their associate…
This paper develops the structure-preserving finite volume weighted essentially non-oscillatory (WENO) hybrid schemes for the shallow water equations under the arbitrary Lagrangian-Eulerian (ALE) framework, dubbed as ALE-WENO schemes. The WENO hybrid reconstruction is adopted on moving meshes, which distinguishes the smooth, non-smooth, and transition stencils by a simple smoothness detector. To maintain the positivity preserving and the well-balanced properties of the ALE-WENO schemes, we adapt the positivity preserving limiter and the well-balanced approaches on static meshes to moving meshe…
In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the "lake equations" for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and …
In our latest studies, by introducing the novel order-preserving (OP) criterion, we have successfully addressed the widely concerned issue of the previously published mapped weighted essentially non-oscillatory (WENO) schemes that it is rather difficult to achieve high resolutions on the premise of removing spurious oscillations for long-run simulations of the hyperbolic systems. In the present study, we extend the OP criterion to the WENO-Z-type schemes as the forementioned issue has also been extensively observed numerically for these schemes. Firstly, we innovatively present the concept of …
On the idea of mapped WENO-JS scheme, properties of mapping methods are analyzed, uncertainties in mapping development are investigated, and new rational mappings are proposed. Based on our former understandings, i.e. mapping at endpoints {0, 1} tending to identity mapping, an integrated Cm,n condition is summarized for function development. Uncertainties, i.e., whether the mapping at endpoints would make mapped scheme behave like WENO or ENO, whether piecewise implementation would entail numerical instability, and whether WENO3-JS could preserve the third-order at first-order critical points …
In this paper, we propose a high order semi-implicit well-balanced finite difference scheme for all Mach Euler equations with a gravitational source term. To obtain the asymptotic preserving property, we start from the conservative form of full compressible Euler equations and add the evolution equation of the perturbation of potential temperature. The resulting system is then split into a (non-stiff) nonlinear low dynamic material wave to be treated explicitly, and (stiff) fast acoustic and gravity waves to be treated implicitly. With the aid of explicit time evolution for the perturbation of…
We develop a positivity-preserving finite difference WENO scheme for the Ten-Moment equations with body forces acting as a source in the momentum and energy equations. A positive forward Euler scheme under a CFL condition is first constructed which is combined with an operator splitting approach together with an integrating factor, strong stability preserving Runge-Kutta scheme. The positivity of the forward Euler scheme is obtained under a CFL condition by using a scaling type limiter, while the solution of the source operator is performed exactly and is positive without any restriction on th…
A new type of finite volume WENO schemes for hyperbolic problems was devised in [36] by introducing the order-preserving (OP) criterion. In this continuing work, we extend the OP criterion to the WENO-Z-type schemes. We firstly rewrite the formulas of the Z-type weights in a uniform form from a mapping perspective inspired by extensive numerical observations. Accrodingly, we build the concept of the locally order-preserving (LOP) mapping which is an extension of the order-preserving (OP) mapping and the resultant improved WENO-Z-type schemes are denoted as LOP-GMWENO-X. There are four major ad…
In this paper, we propose to combine the fifth order Hermite weighted essentially non-oscillatory (HWENO) scheme and fast sweeping method (FSM) for the solution of the steady-state $S_{N}$ transport equation in the finite volume framework. It is well-known that the $S_{N}$ transport equation asymptotically converges to a macroscopic diffusion equation in the limit of optically thick systems with small absorption and sources. Numerical methods which can preserve the asymptotic limit are referred to as asymptotic preserving methods. In the one-dimensional case, we provide the analysis to demonst…
A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry
TU Kaiserslautern（凯泽斯劳滕工业大学）；RWTH Aachen University（亚琛工业大学）
In this paper, a high order free-stream preserving finite difference weighted essentially non-oscillatory (WENO) scheme is developed for the ideal magnetohydrodynamic (MHD) equations on curvilinear meshes. Under the constrained transport framework, magnetic potential evolved by a Hamilton-Jacobi (H-J) equation is introduced to control the divergence error. In this work, we use the alternative formulation of WENO scheme [10] for the nonlinear hyperbolic conservation law, and design a novel method to solve the magnetic potential. Theoretical derivation and numerical results show that the scheme …
In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the positivity-preserving property of mPDeC, the resulting scheme is unconditionally positivity preserving for the water height. To apply the mPDeC approach, we have to interpret the spatial semi-discretization in terms of production-destruction systems. Only small modifications inside the classical WENO implementation are necessary and we explain how it can be done. …
This paper presents a highly robust third-order accurate finite volume weighted essentially non-oscillatory (WENO) method for special relativistic hydrodynamics on unstructured triangular meshes. We rigorously prove that the proposed method is physical-constraint-preserving (PCP), namely, always preserves the positivity of the pressure and the rest-mass density as well as the subluminal constraint on the fluid velocity. The method is built on a highly efficient compact WENO reconstruction on unstructured meshes, a simple PCP limiter, the provably PCP property of the Harten--Lax--van Leer flux,…
We develop new adaptive alternative weighted essentially non-oscillatory (A-WENO) schemes for hyperbolic systems of conservation laws. The new schemes employ the recently proposed local characteristic decomposition based central-upwind numerical fluxes, the three-stage third-order strong stability preserving Runge-Kutta time integrator, and the fifth-order WENO-Z interpolation. The adaptive strategy is implemented by applying the limited interpolation only in the parts of the computational domain where the solution is identified as ``rough'' with the help of a smoothness indicator. We develop …
In this paper, we propose a new high order semi-implicit scheme for the all Mach full Euler equations of gas dynamics. Material waves are treated explicitly, while acoustic waves are treated implicitly, thus avoiding severe CFL restrictions for low Mach flows. High order accuracy in time is obtained by semi-implicit temporal integrator based on the IMEX Runge-Kutta (IMEX-RK) framework. High order in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions adapted to the semi-implicit IMEX-RK time discretization. Type A IMEX schemes are constructed to handle …
Provably convergent Newton-Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics
This paper develops the genuinely multidimensional HLL Riemann solver for the two-dimensional special relativistic hydrodynamic equations on Cartesian meshes and studies its physical-constraint preserving (PCP) property. Based on the resulting HLL solver, the first- and high-order accurate PCP finite volume schemes are proposed. In the high-order scheme, the WENO reconstruction, the third-order accurate strong-stability-preserving time discretizations and the PCP flux limiter are used. Several numerical results are given to demonstrate the accuracy, performance and resolution of the shock wave…
We present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov-Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. Fourth-order accuracy is accomplished in both space and time under a non-splitting setting. Mass …
The weighted essentially non-oscillatory (WENO) methods are popular and effective spatial discretization methods for nonlinear hyperbolic partial differential equations. Although these methods are formally first-order accurate when a shock is present, they still have uniform high-order accuracy right up to the shock location. In this paper, we propose a novel third-order numerical method for solving optimal control problems subject to scalar nonlinear hyperbolic conservation laws. It is based on the first-disretize-then-optimize approach and combines a discrete adjoint WENO scheme of third ord…
