崔雪微

Kuramoto-Tsuzuki方程一階線性向后歐拉有限元方法的最優(yōu)誤差估計

崔雪微

(溫州大學數(shù)理學院，浙江溫州 325035)

對于高維非線性Kuramoto-Tsuzuki方程，給出了一階向后歐拉有限元全離散格式，并對非線性項采用半隱格式，從理論上證明了離散解的穩(wěn)定性以及離散解與精確解的無條件最優(yōu)誤差估計．

一階線性向后歐拉有限元方法；無條件最優(yōu)化誤差估計；Kuramoto-Tsuzuki方程；高維非線性問題

Kuramoto-Tsuzuki方程用于描述分叉點附近的兩個系統(tǒng)的行為[1-2]，它也可以看作是復雜的Ginzburg-Landua方程的一個特例，該方程被廣泛用于描述從非線性波到二階相變，從超導和玻色-愛因斯坦凝聚到場論中的液晶和弦的大量現(xiàn)象①Feb. 2022 Kuramoto Y. Chemical Oscillations, Waves, and Turbulence [M]. Berlin: Springer Verlag, 1984.等．

本文主要考慮如下的高維Kuramoto-Tsuzuki方程：

本文建立Kuramoto-Tsuzuki方程的一階向后歐拉有限元全離散格式，以此證明離散解與精確解的無條件最優(yōu)誤差估計．

1 預備知識

由經(jīng)典有限元理論可知有：

和

成立．

2 主要結(jié)果

對于一階線性向后歐拉有限元解，有如下的穩(wěn)定性結(jié)論．

考慮方程（8）的實部，有：

即有：

因此有：

引理2得證．

3 主要結(jié)果的證明

用（11）式減去（7）式，并應(yīng)用Ritz投影定義，可得：

考慮方程（13）的實部，并應(yīng)用Cauchy-Schwarz不等式可得：

由（2）式及Taylor展式可得：

由Young不等式及（2）式可得：

由逆不等式及Young不等式可得：

把（21）―（24）式代入（20）式，可得：

同理可得：

把（18）式、（19）式、（25）式和（26）式代入（17）式中可得：

把（15）式、（16）式和（27）式代入（11）式中，可得：

由逆不等式（6）和Sobolev嵌入定理（4）可得：

由（4）式和（5）式可得：

整理后可得：

定理1得證．

[1] Kuramoto Y, Tsuzuki T. On the Formation of Dissipative Structures in Reaction-diffusion Systems [J]. Progress of Theoretical Physics, 1975, 54(3): 678-699.

[2] Akhromeeva T S, Kurdyumov S P, Malinetskii G G, et al. Classification of Solutions of a System of Nonlinear Diffusion Equations in a Neighborhood of a Bifurcation Point [J]. Journal of Soviet Mathematics, 1988, 41(5): 1292-1356.

[3] Tsertsvadze G Z. On the Convergence of Difference Schemes for the Kuramoto-Tsuzuki Equation and Reaction- diffusion Type Systems [J]. Computational Mathematics & Mathematical Physics, 1992, 31(5): 40-47.

[4] Sun Z Z. On Tsertsvadze’s Difference Scheme for the Kuramoto-Tsuzuki Equation [J]. Journal of Computational Applied Mathematics, 1998, 98(2): 289-304.

[5] Omrani K. Convergence of Galerkin Approximations for the Kuramoto-Tsuzuki Equation [J]. Numerical Methods for Partial Differential Equations, 2005, 21(5): 961-975.

[6] Wang S, Wang T, Zhang L, et al. Convergence of a Nonlinear Finite Difference Scheme for the Kuramoto-Tsuzuki Equation [J]. Communications in Nonlinear Science Numerical Simulation, 2011, 16(6): 2620-2627.

[7] Sun Z Z. A Linear Difference Scheme for the Kuramoto-Tsuzuki Equation [J]. Journal of Computational Mathematics, 1996, 14(1): 1-7.

[8] 孫志忠. 數(shù)值求解Kuramoto-Tsuzuki方程的廣義Box格式[J]. 東南大學學報, 1996, 26(1): 87-92.

[9] Wang T, Guo B. A Robust Semi-explicit Difference Scheme for the Kuramoto-Tsuzuki Equation [J]. Journal of Computational and Applied Mathematics, 2009, 233(4): 878-888.

[10] Hu X, Chen S, Chang Q. Fourth-order Compact Difference Scheme for 1D Nonlinear Kuramoto-Tsuzuki Equation [J]. Numerical Methods for Partial Differential Equations, 2015, 31(6): 2080-2109.

[11] Leonaviciene T, Bugajev A, Jankeviciute G, et al. On Stability Analysis of Finite Difference Schemes for Generalized Kuramoto-Tsuzuki Equation with Nonlocal Boundary Conditions [J]. Mathematical Modelling and Analysis, 2016, 21(5): 630-643.

[12] Li B, Sun W. Error Analysis of Linearized Semi-implicit Galerkin Finite Element Methods for Nonlinear Parabolic Equations [J]. International Journal of Numerical Analysis Modeling, 2013, 10(3): 622-633.

[13] Li B, Sun W. Unconditional Convergence and Optimal Error Estimates of a Galerkin-mixed FEM for Incompressible Miscible Flow in Porous Media [J]. SIAM Journal on Numerical Analysis, 2013, 51(4): 1959-1977.

[14] Li D F. Optimal Error Estimates of a Linearized Crank-Nicolson Galerkin FEM for the Kuramoto-Tsuzuki Equations [J]. Communications in Computational Physics, 2019, 26(3): 838-854.

Optimal Error Estimates of a First-order Linearized Backward Euler FEM for the Kuramoto-Tsuzuki Equations

CUI Xuewei

(College of Mathematics and Physics, Wenzhou University, Wenzhou, China 325035)

For the high-dimensional nonlinear Kuramo-Tsuzuki equations, a first-order backward Euler finite factor is given in a fully discrete format. The semi-hidden format of nonlinear terms is used, which theoretically proves the stability of discrete solution and the unconditional optimal error estimates of discrete solution and precise solution.

First-order Backward Euler FEM; Unconditional Optimal Error Estimates; Kuramo-Tsuzuki Equation; High Dimensional Nonlinear Problem

O241.82

A

1674-3563(2022)01-0017-08

10.3875/j.issn.1674-3563.2022.01.003

本文的PDF文件可以從www.wzu.edu.cn/wzdxxb.htm獲得

2020-11-26

崔雪微(1993―)，女，黑龍江綏化人，碩士研究生，研究方向：偏微分方程數(shù)值解

(編輯：封毅)

(英文審校：黃璐)