Superconvergence Analysis of the Runge–Kutta Discontinuous Galerkin Methods for a Linear Hyperbolic Equation

Yuan, Xu; Meng, Xiong; Shu, Chi‐Wang; Zhang, Qiang

doi:10.1007/s10915-020-01274-1

Cited by 19 publications

(14 citation statements)

References 34 publications

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“…Another stream of studies particularly focuses on discontinuous Galerkin (DG) operators for linear conservation laws. Besides stability results that are consistent with those for general operators, it is also shown that the methods can achieve strong stability with extra low order spatial polynomials [53,39,50,51]. For nonlinear problems, counterexamples were given by Ranocha in [32] showing that various explicit RK methods do not preserve strong stability and order barriers were proved for certain strong-stability-preserving RK methods.…”

Section: Introductionmentioning

confidence: 52%

Enforcing strong stability of explicit Runge--Kutta methods with superviscosity

Sun

Shu

2019

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A time discretization method is called strongly stable, if the norm of its numerical solution is nonincreasing. It is known that, even for linear semi-negative problems, many explicit Runge-Kutta (RK) methods fail to preserve this property. In this paper, we enforce strong stability by modifying the method with superviscosity, which is a numerical technique commonly used in spectral methods. We propose two approaches, the modified method and the filtering method for stabilization. The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term; the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity. For linear problems, most explicit RK methods can be stabilized with either approach without accuracy degeneration. Furthermore, we prove a sharp bound (up to an equal sign) on diffusive superviscosity for ensuring strong stability. The bound we derived for general dispersive-diffusive superviscosity is also verified to be sharp numerically. For nonlinear problems, a filtering method is investigated for stabilization. Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximation of conservation laws are performed to validate our analysis and to test the performance.

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Section: Introductionmentioning

confidence: 52%

Enforcing strong stability of explicit Runge--Kutta methods with superviscosity

Sun

Shu

2019

Preprint

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“…The stability analyses in Theorem 3.7 and [35] are closely connected with the L 2 -stability analysis of RK discontinuous Galerkin schemes for the linear advection equation by Xu et al in [43,42], where the weak(κ) stability was systematically studied, and the property u n+1 2 ≤ u n 2 was called monotonicity stability in [43,42]. The discussions in [43,35,42] were focused on explicit RK methods. In the present paper, our framework, including the discrete energy laws and stability results, applies to both general implicit and explicit RK methods.…”

Section: Stability Analysismentioning

confidence: 96%

“…In particular, it was proved in [35] that all linear RK methods corresponding to pth order truncated Taylor expansions are strongly stable if p ≡ 3 (mod 4) and are not strongly stable if p ≡ 1 or 2 (mod 4). It is worth noting that the stability analysis in [35] is closely related to that of the RK discontinuous Galerkin schemes for linear advection equation by Xu et al in [43,42]. For nonlinear or nonautonomous problems, the requirement for strong stability may lead to order barriers [28,29].…”

mentioning

confidence: 96%

“…The discrete energy laws are important and helpful for further understanding the stability of RK methods, which is a classical topic in numerical analysis. Over the past decades, rich mathematical theories on the stability of RK methods have been developed both in the ODE settings (see [39, Chapter IV], [5,Chapter 3], and references therein) and in the context of numerical PDEs (see [3,44,8,38,43,42,9] and references therein).…”

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confidence: 99%

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On Energy Laws and Stability of Runge--Kutta Methods for Linear Seminegative Problems

Sun¹,

Wei²,

K³

2022

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This paper presents a systematic theoretical framework to derive the energy identities of general implicit and explicit Runge-Kutta (RK) methods for linear seminegative systems. It generalizes the stability analysis of explicit RK methods in [Z. Sun and C.-W. Shu, SIAM J. Numer. Anal., 57 (2019), pp. 1158-1182. The established energy identities provide a precise characterization on whether and how the energy dissipates in the RK discretization, thereby leading to weak and strong stability criteria of RK methods. Furthermore, we discover a unified energy identity for all the diagonal Padé approximations, based on an analytical Cholesky type decomposition of a class of symmetric matrices. The structure of the matrices is very complicated, rendering the discovery of the unified energy identity and the proof of the decomposition highly challenging. Our proofs involve the construction of technical combinatorial identities and novel techniques from the theory of hypergeometric series. Our framework is motivated by a discrete analogue of integration by parts technique and a series expansion of the continuous energy law. In some special cases, our analyses establish a close connection between the continuous and discrete energy laws, enhancing our understanding of their intrinsic mechanisms. Several specific examples of implicit methods are given to illustrate the discrete energy laws. A few numerical examples further confirm the theoretical properties.

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“…Many papers have used this technique to show the semi-discrete DG method's superconvergence for one-dimensional problems. One of the recent papers done by Xu, Meng, Shu, Zhang [24] uses a slightly modified correction function and the L2-norm stability to establish the superconvergence property of the Runge-Kutta discontinuous Galerkin method for solving a linear constant-coefficient hyperbolic equation. They show that under a r+1 temporal and 2k+2 spatial smoothness assumption, and by choosing a specific initialization, the cell average and the numerical flux are min(2k + 1, r) superconvergent.…”

Section: Introductionmentioning

confidence: 99%

Superconvergence of discontinuous Galerkin method for scalar and vector linear advection equations

Rahmati¹,

Lü²

2021

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In this paper, we use Fourier analysis to study the superconvergence of the semi-discrete discontinuous Galerkin method for scalar linear advection equations in one spatial dimension. The error bounds and asymptotic errors are derived for initial discretization by L2 projection, Gauss-Radau projection, and other projections proposed by Cao et. al. [1]. For pedagogical purpose, the errors are computed in two different ways. In the first approach, we compute the difference between the numerical solution and a special interpolation of the exact solution, and show that it consists of an asymptotic error of order 2k + 1 and a transient error of lower order. In the second approach, as in Ref.[2], we compute the error directly by decomposition into physical and nonphysical modes, and obtain agreement with the first approach. We then extend the analysis to vector conservation laws, solved using the Lax-Friedrichs flux. We prove that the superconvergence holds with the same order. The error bounds and asymptotic errors are demonstrated by various numerical experiments for scalar and vector advection equations.

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Superconvergence Analysis of the Runge–Kutta Discontinuous Galerkin Methods for a Linear Hyperbolic Equation

Cited by 19 publications

References 34 publications

Enforcing strong stability of explicit Runge--Kutta methods with superviscosity

Enforcing strong stability of explicit Runge--Kutta methods with superviscosity

On Energy Laws and Stability of Runge--Kutta Methods for Linear Seminegative Problems

Superconvergence of discontinuous Galerkin method for scalar and vector linear advection equations

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