In this work, we consider an $ \alpha $-robust high-order numerical method for the time fractional nonlinear Korteweg-de Vries (KdV) equation. The time fractional derivatives are discretized by the L1 formula based on the graded meshes. For the spatial derivative, the nonlinear operator is defined to approximate the $ uu_x $, and two coupling equations are obtained by processing the $ u_{xxx} $ with the order reduction method. Finally, the nonlinear difference schemes with order ($ 2-\alpha $) in time and order $ 2 $ precision in space are obtained. This means that we can get a higher precision solution and improve the computational efficiency. The existence and uniqueness of numerical solutions for the proposed nonlinear difference scheme are proved theoretically. It is worth noting the unconditional stability and $ \alpha $-robust stability are also derived. Moreover, the optimal convergence result in the $ L_2 $ norms is attained. Finally, two numerical examples are given, which is consistent with the theoretical analysis.
Citation: Caojie Li, Haixiang Zhang, Xuehua Yang. A new $ \alpha $-robust nonlinear numerical algorithm for the time fractional nonlinear KdV equation[J]. Communications in Analysis and Mechanics, 2024, 16(1): 147-168. doi: 10.3934/cam.2024007
In this work, we consider an $ \alpha $-robust high-order numerical method for the time fractional nonlinear Korteweg-de Vries (KdV) equation. The time fractional derivatives are discretized by the L1 formula based on the graded meshes. For the spatial derivative, the nonlinear operator is defined to approximate the $ uu_x $, and two coupling equations are obtained by processing the $ u_{xxx} $ with the order reduction method. Finally, the nonlinear difference schemes with order ($ 2-\alpha $) in time and order $ 2 $ precision in space are obtained. This means that we can get a higher precision solution and improve the computational efficiency. The existence and uniqueness of numerical solutions for the proposed nonlinear difference scheme are proved theoretically. It is worth noting the unconditional stability and $ \alpha $-robust stability are also derived. Moreover, the optimal convergence result in the $ L_2 $ norms is attained. Finally, two numerical examples are given, which is consistent with the theoretical analysis.
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Caojie Li, Haixiang Zhang, Xuehua Yang. A new $ \alpha $-robust nonlinear numerical algorithm for the time fractional nonlinear KdV equation[J]. Communications in Analysis and Mechanics, 2024, 16(1): 147-168. doi: 10.3934/cam.2024007
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