Sixth-order Weighted Essentially Nonoscillatory Schemes Based on Exponential Polynomials

Ha, Youngsoo; Kim, Chang Ho; Yang, Hyoseon; Yoon, Jungho

doi:10.1137/15m1042814

Cited by 18 publications

(18 citation statements)

References 32 publications

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“…In [7,8], the notion of the generalized undivided difference to higher order functionals was introduced so that it can be used to achieve a higher order approximation to the derivatives of a function than the classical undivided difference. We will apply the generalized undivided difference to the substencils S k , k = 0, 1, 2, for the L 1 -norm smoothness indicators.…”

Section: New Nonlinear Weights Based On Generalized Undivided Differe...mentioning

confidence: 99%

“…One is the extension of the MWENO nonlinear weights, which are based on the smoothness indicators in [15] of L 2 norms, to a more general Z-type nonlinear weights. The other is to design the novel Z-type nonlinear weights depending on new smoothness indicators with the notion of generalized undivided difference [8]. We take full advantage of the information about the derivatives of a flux function on a given substencil.…”

Section: Introductionmentioning

confidence: 99%

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Sixth order weighted essentially non-oscillatory schemes with Z-type nonlinear weighting procedure for nonlinear degenerate parabolic equations

Rathan¹,

Gu²

2022

Preprint

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In this paper we develop new nonlinear weights of sixth order finite difference weighted essentially non-oscillatory (WENO) schemes for nonlinear degenerate parabolic equations. We construct two Z-type nonlinear weights: one is based on the L 2 norm and the other depends on the L 1 norm, yielding improved WENO schemes with more accurate resolution. We also confirm that the new devised nonlinear weights satisfy the sufficient conditions of sixth order accuracy. Finally, one-and two-dimensional numerical examples are presented to demonstrate the improved behavior of WENO schemes with new weighting procedure.

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Section: New Nonlinear Weights Based On Generalized Undivided Differe...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sixth order weighted essentially non-oscillatory schemes with Z-type nonlinear weighting procedure for nonlinear degenerate parabolic equations

Rathan¹,

Gu²

2022

Preprint

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“…The process to approximate J L [v; α](x i ) is simply mirror symmetric to that of J R [v; α](x i ) with respect to point x i , so we will illustrate the process only for the term J R [v; α](x i ). In [18], the authors introduced a sixth-order WENO scheme based on the exponential polynomial space, which we shall follow for approximating J R [v; α](x i ). To begin, we consider an interpolation stencil consisting of k + 1 points, which contains x i−1 and x i :…”

Section: Space Discretization With Exponential Based Weno Schemesmentioning

confidence: 99%

“…for the four-point substencils. Here Γ 6 and Γ 4 constitute extended Tchebysheff systems on R so that the non-singularity of the interpolation matrices in (3.12) is guaranteed, see [18,21].…”

Section: Space Discretization With Exponential Based Weno Schemesmentioning

confidence: 99%

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A kernel based high order “explicit” unconditionally stable scheme for time dependent Hamilton–Jacobi equations

Christlieb

Guo

Jiang

2019

Journal of Computational Physics

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In [11], the authors developed a class of high-order numerical schemes for the Hamilton-Jacobi (H-J) equations, which are unconditionally stable, yet take the form of an explicit scheme. This paper extends such schemes, so that they are more effective at capturing sharp gradients, especially on nonuniform meshes. In particular, we modify the weighted essentially non-oscillatory (WENO) methodology in the previously developed schemes by incorporating an exponential basis and adapting the previously developed nonlinear filters used to control oscillations. The main advantages of the proposed schemes are their effectiveness and simplicity, since they can be easily implemented on higher-dimensional nonuniform meshes. We perform numerical experiments on a collection of examples, including H-J equations with linear, nonlinear, convex and non-convex Hamiltonians. To demonstrate the flexibility of the proposed schemes, we also include test problems defined on non-trivial geometry.Recent work on the MOL T has involved extending the method to solve more general nonlinear PDEs, for which an integral solution is generally not applicable. This work includes the nonlinear degenerate convection-diffusion equations [10], as well as the H-J equations [11]. The key idea of these papers involved exploiting the linearity of a given differential operator, rather than requiring linearity in the underlying equations. This allowed derivative operators in the problems to be expressed through kernel representations developed for linear problems. Formulating applicable derivative operators in this way ultimately facilitated the stability of the schemes, since a global coupling was introduced through the integral operator. As part of this embedding process, a kernel parameter β was introduced, and through a careful selection, was shown to yield schemes which are A-stable. Remarkably, it was shown that one could couple these representations for the derivative operators with an explicit time-stepping method, such as the strong-stability-preserving Runge-Kutta (SSP-RK) methods [17] and still obtain schemes which maintain unconditional stability [10,11]. To address shock-capturing and control non-physical oscillations, the latter two papers introduced quadrature formulas based on WENO reconstruction, along with a nonlinear filter.This paper seeks to extend the work in [10,11] to the H-J equations (1.1) defined on non-uniformly distributed spatial domains. In particular, several improvements are given. First, we develop the MOL T for mapped grids using a general coordinate transformation function, which allows for a non-uniform distribution of grid points. We show that, with this mapping, our numerical scheme is able to preserve the conservation property for the derivative of the solution to the H-J equation. We also describe a novel WENO-based quadrature for the spatial discretization, which uses a basis consisting of exponential polynomials, to improve the shock capturing capabilities of the method. Another difference in this paper, compared to...

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Exponential approximation space reconstruction weighted essentially nonoscillatory scheme for dispersive partial differential equations

Salian,

Samala

2023

Math Methods in App Sciences

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In this work, we construct a fifth‐order weighted essentially non‐oscillatory (WENO) scheme with exponential approximation space for solving dispersive equations. A conservative third‐order derivative formulation is developed directly using WENO spatial reconstruction procedure, and third‐order TVD Runge–Kutta scheme is used for the evaluation of time derivative. This exponential approximation space consists a tension parameter that may be optimized to fit the specific feature of the characteristic data, yielding better results without spurious oscillations compared to the polynomial approximation space. A detailed formulation is presented for the construction of conservative flux approximation, smoothness indicators, nonlinear weights, and verified that the proposed scheme provides the required fifth convergence order. One‐ and two‐dimensional numerical examples are presented to support the theoretical claims.

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Sixth-order Weighted Essentially Nonoscillatory Schemes Based on Exponential Polynomials

Cited by 18 publications

References 32 publications

Sixth order weighted essentially non-oscillatory schemes with Z-type nonlinear weighting procedure for nonlinear degenerate parabolic equations

Sixth order weighted essentially non-oscillatory schemes with Z-type nonlinear weighting procedure for nonlinear degenerate parabolic equations

A kernel based high order “explicit” unconditionally stable scheme for time dependent Hamilton–Jacobi equations

Exponential approximation space reconstruction weighted essentially nonoscillatory scheme for dispersive partial differential equations

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