A characteristics-mixed finite element method for Burgers’ equation

Chen, Huanzhen; Jiang, Ziwen

doi:10.1007/bf02935745

Cited by 15 publications

(9 citation statements)

References 11 publications

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“…Moreover, when

u_{0} \in H^{1} (E),

we have the following result . Lemma Suppose that

\partial E \in C^{2},

and

u_{0} \in H^{1} (E)

.…”

Section: The Crank–nicolson–galerkin Scheme For Solving the Burgers' mentioning

confidence: 92%

Multilevel augmentation methods for solving the Burgers' equation

Chen

Cheng

et al. 2015

Numer. Methods Partial Differential Eq.

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In this article, multilevel augmentation method (MAM) for solving the Burgers' equation is developed. The Crank-Nicolson-Galerkin scheme of the Burgers' equation results in nonlinear algebraic systems at each time step, the computational cost for solving these nonlinear systems is huge. The MAM allows us to solve the nonlinear system at a fixed initial lower level and then compensate the error by solving a linear system at the higher level. We prove that the method has the same optimal convergence order as the projection method, while reducing the computational complexity greatly. Finally, numerical experiments are presented to confirm the theoretical analysis and illustrate the efficiency of the proposed method.

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“…Moreover, when

u_{0} \in H^{1} (E),

we have the following result . Lemma Suppose that

\partial E \in C^{2},

and

u_{0} \in H^{1} (E)

.…”

Section: The Crank–nicolson–galerkin Scheme For Solving the Burgers' mentioning

confidence: 92%

Multilevel augmentation methods for solving the Burgers' equation

Chen

Cheng

et al. 2015

Numer. Methods Partial Differential Eq.

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“…If one would pretend to deal with the case where discontinuous solutions were expected, appropriate shock-capturing schemes should be chosen. We refer to [10,13,22] and [32] or to the references therein for such type of approaches. Such case should be subject of future research.…”

Section: Problem and Semi-discretizationmentioning

confidence: 99%

Numerical simulations for the stabilization and estimation problem of a semilinear partial differential equation

Tiago

2015

Applied Numerical Mathematics

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“…Luo and Liu [14] proposed an MFE method to solve one-dimensional Burgers' equation by introducing a flux function as an auxiliary variable and gave the existence, uniqueness, and error analyses for the discrete solutions. Chen and Jiang [15] constructed a characteristic MFE scheme to solve onedimensional Burgers' equation and obtained the optimal error estimates for the velocity and flux (gradient) in the L 2 norm. Pany et al [16] applied an H 1 -Galerkin MFE method to approximate the velocity and flux of one-dimensional Burgers' equation and gave a priori error estimates and numerical experiments.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Burgers’ Equation Based on Mixed Finite Volume Element Methods

Zhao

Fang

et al. 2020

Discrete Dynamics in Nature and Society

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In this article, mixed finite volume element (MFVE) methods are proposed for solving the numerical solution of Burgers’ equation. By introducing a transfer operator, semidiscrete and fully discrete MFVE schemes are constructed. The existence, uniqueness, and stability analyses for semidiscrete and fully discrete MFVE schemes are given in detail. The optimal a priori error estimates for the unknown and auxiliary variables in the L2Ω norm are derived by using the stability results. Finally, numerical results are given to verify the feasibility and effectiveness.

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A characteristics-mixed finite element method for Burgers’ equation

Cited by 15 publications

References 11 publications

Multilevel augmentation methods for solving the Burgers' equation

Multilevel augmentation methods for solving the Burgers' equation

Numerical simulations for the stabilization and estimation problem of a semilinear partial differential equation

Numerical Solution of Burgers’ Equation Based on Mixed Finite Volume Element Methods

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