Order Finite Difference Weighted Essentially Non-oscillatory Schemes 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.

A new fifth order finite difference WENO scheme to improve convergence rate at critical points

In the present work, we construct a new, improved version of fifth-order finite-difference Weighted Essentially Non-Oscillatory (WENO) scheme with less dissipation to approximate solutions for one- and two-dimensional hyperbolic conservation laws and associated problems. The higher-order information presented for the classical WENO scheme is applied to construct a new global smoothness indicator, a composition of first derivatives of local smoothness indicators with their respective interpolation polynomials. The sufficient condition of nonlinear weights is satisfied by using Taylor expansions. We verify the proposed scheme through several benchmark test cases for scalar and system of hyperbolic conservation laws. We observe from the numerical simulations that the designed WENO scheme demonstrates high resolutions, fifth-order accuracy, and robustness at critical points as compared with other modern WENO schemes.

A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws

In this paper a new simple fifth order weighted essentially non-oscillatory (WENO) scheme is presented in the finite difference framework for solving the hyperbolic conservation laws. The new WENO scheme is a convex combination of a fourth degree polynomial with two linear polynomials in a traditional WENO fashion. This new fifth order WENO scheme uses the same five-point information as the classical fifth order WENO scheme 14,20, could get less absolute truncation errors in L 1 and L ∞ norms, and obtain the same accuracy order in smooth region containing complicated numerical solution structures simultaneously escaping nonphysical oscillations adjacent strong shocks or contact discontinuities. The associated linear weights are artificially set to be any random positive numbers with the only requirement that their sum equals one. New nonlinear weights are proposed for the purpose of sustaining the optimal fifth order accuracy. The new WENO scheme has advantages over the classical WENO scheme 14,20 in its simplicity and easy extension to higher dimensions. Some benchmark numerical tests are performed to illustrate the capability of this new fifth order WENO scheme.

A high-order weighted essentially non-oscillatory (WENO) finite difference scheme for nonlinear degenerate parabolic equations

In this paper, we propose a new WENO finite difference procedure for nonlinear degenerate parabolic equations which may contain discontinuous solutions. Our scheme is based on the method of lines, with a high-order accurate conservative approximation to each of the diffusion terms based on an idea that has been recently presented by Liu et al. [Y. Liu, C.-W. Shu, M. Zhang, High order finite difference WENO schemes for non-linear degenerate parabolic equations, SIAM J. Sci. Comput. 33 (2011) 939–965]. Our scheme tries to circumvent the negative ideal weights that appear when applying the standard WENO idea, as is done in Liu et al. (2011) [13]. In one-dimensional case, first we obtain an optimum polynomial on a six-points stencil. This optimum polynomial is sixth-order accurate in regions of smoothness. Then, we consider this optimum polynomial as a symmetric and convex combination of four polynomials with ideal weights. Following the methodology of the classic WENO procedure, then we calculate the non-oscillatory weights with the ideal weights. Numerical examples are provided to demonstrate the resolution power and accuracy of the scheme. Finally, the new method is extended to multi-dimensional problems by dimension-by-dimension approach. More examples of multi-dimension problems are presented to show that our method remains non-oscillatory while giving good resolution of discontinuities. Finally, we would like to mention that this paper combines and extends the techniques proposed in [13] and Levy et al. (2000) [24].

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 ...

High order finite difference WENO schemes for fractional differential equations

This letter develops high order finite difference weighted essentially non-oscillatory (WENO) schemes for fractional differential equations. First, the αth, 1<α≤2, Caputo fractional derivative is split into a classical second derivative and a weakly singular integral. Then the sixth-order finite difference WENO scheme is used to discretize the classical second derivative and the Gauss–Jacobi quadrature is applied to solve the weakly singular integral. The constructed scheme of approximation for the fractional derivative has high order accuracy in smooth regions and maintains a sharp discontinuity transition. Finally, numerical experiments are performed to demonstrate the effectiveness of the proposed schemes.

Positivity-preserving high order finite difference WENO schemes for compressible Euler equations

In Zhang and Shu (2010) [20], Zhang and Shu (2011) [21] and Zhang et al. (in press) [23], we constructed uniformly high order accurate discontinuous Galerkin (DG) and finite volume schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this framework to construct positivity-preserving high order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) finite difference schemes for compressible Euler equations. General equations of state and source terms are also discussed. Numerical tests of the fifth order finite difference WENO scheme are reported to demonstrate the good behavior of such schemes.

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.

High Order Finite Difference WENO Schemes for Nonlinear Degenerate Parabolic Equations

High order accurate weighted essentially nonoscillatory (WENO) schemes are usually designed to solve hyperbolic conservation laws or to discretize the first derivative convection terms in convection dominated partial differential equations. In this paper we discuss a high order WENO finite difference discretization for nonlinear degenerate parabolic equations which may contain discontinuous solutions. A porous medium equation (PME) is used as an example to demonstrate the algorithm structure and performance. By directly approximating the second derivative term using a conservative flux difference, the sixth order and eighth order finite difference WENO schemes are constructed. Numerical examples are provided to demonstrate the accuracy and nonoscillatory performance of these schemes.

Anti-diffusive flux corrections for high order finite difference WENO schemes

In this paper, we generalize a technique of anti-diffusive flux corrections, recently introduced by Després and Lagoutière [Journal of Scientific Computing 16 (2001) 479–524] for first-order schemes, to high order finite difference weighted essentially non-oscillatory (WENO) schemes. The objective is to obtain sharp resolution for contact discontinuities, close to the quality of discrete traveling waves which do not smear progressively for longer time, while maintaining high order accuracy in smooth regions and non-oscillatory property for discontinuities. Numerical examples for one and two space dimensional scalar problems and systems demonstrate the good quality of this flux correction. High order accuracy is maintained and contact discontinuities are sharpened significantly compared with the original WENO schemes on the same meshes.

A Hybrid Cosmological Hydrodynamic/N‐Body Code Based on a Weighted Essentially Nonoscillatory Scheme

We describe a newly developed cosmological hydrodynamics code based on the weighted essentially non-oscillatory (WENO) schemes for hyperbolic conservation laws. High order finite difference WENO schemes are designed for problems with piecewise smooth solutions containing discontinuities, and have been successful in applications for problems involving both shocks and complicated smooth solution structures. We couple hydrodynamics based on the WENO scheme with standard Poisson solver - particle-mesh (PM) algorithm for evolving the self-gravitating system. A third order total variation diminishing (TVD) Runge-Kutta scheme has been used for time-integration of the system. We brief the implementation of numerical technique. The cosmological applications in simulating intergalactic medium and Ly$\alpha$ forest in the CDM scenario are also presented.