An implicit robust numerical scheme with graded meshes for the modified Burgers model with nonlocal dynamic properties
An implicit robust numerical scheme with graded meshes for the modified Burgers model with nonlocal dynamic properties
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DOI：
10.1007/s40314-023-02373-z
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发表时间：
2023-07
影响因子：
2.6
通讯作者：
Qingqing Tian;Xuehua Yang;Haixiang Zhang;Da Xu
中科院分区：
文献类型：
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作者：
Qingqing Tian;Xuehua Yang;Haixiang Zhang;Da Xu
针对具有非局部动力特性的修正Burgers模型，构造了一种具有梯度网格的隐式稳健差分方法。在Caputo意义下的分数阶导数的梯度网格上的L1公式。采用基于分段线性检验函数的Galerkin方法隐式处理非线性对流项，得到非线性代数方程组。对于四阶项和二阶项，采用带整数余项的泰勒展开式。然后证明了该隐式差分格式数值解的存在唯一性。同时，也得到了无条件稳定性.通过引入一个新的离散Gronwall不等式，我们将-稳定性改进为-鲁棒稳定性，即当，界不会爆破。并且给出了在能量范数下的最优收敛阶.最后，我们给出了三个数值实验来说明和比较所提出的鲁棒方法在均匀网格和梯度网格上的效率。结果表明，所得结果与理论分析一致。
In this paper, an implicit robust difference method with graded meshes is constructed for the modified Burgers model with nonlocal dynamic properties. The L1 formula on graded meshes for the fractional derivative in the Caputo sense is employed. The Galerkin method based on piece-wise linear test functions is used to handle the nonlinear convection termimplicitly and attain a system of nonlinear algebraic equations. The Taylor expansion with integral remainder is used to deal with the fourth-order termand the second-order term. Then the existence and uniqueness of numerical solutions for the proposed implicit difference scheme are proved. Meanwhile, the unconditional stability is also derived. By introducing a new discrete Gronwall inequality, we improve the-stability to the-robust stability, that is, when, the bound will not blow up. And we also yield the optimal convergence order in theenergy norm. Finally, we give three numerical experiments to illustrate and compare the efficiency of the proposed robust method on uniform meshes and graded meshes. It shows that the results are consistent with our theoretical analysis.