Abstract To discretize the distributed-order term of two-dimensional nonlinear Riesz space fractional diffusion equation, we consider the high accuracy Gauss-Legendre quadrature formula. By combining an s-stage implicit Runge-Kutta method in temporal direction with a spectral Galerkin method in spatial direction, we construct a numerical method with high global accuracy. If the nonlinear function satisfies the local Lipschitz condition, the s-stage implicit Runge-Kutta method with order p ( p ≥ s + 1 ) is coercive and algebraically stable, then we can prove that the proposed method is stable and convergent of order s + 1 in time. In addition, we also derive the optimal error estimate for the discretization of distributed-order term and spatial term. Finally, numerical experiments are presented to verify the theoretical results.

中文翻译：

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二维非线性 Riesz 空间分布阶扩散方程的隐式 Runge-Kutta 和谱 Galerkin 方法

摘要 为了离散化二维非线性Riesz空间分数阶扩散方程的分布阶项，我们考虑了高精度Gauss-Legendre求积公式。通过将时间方向上的 s 级隐式龙格-库塔方法与空间方向上的谱伽辽金方法相结合，我们构建了一种具有高全局精度的数值方法。如果非线性函数满足局部 Lipschitz 条件，则 p 阶 ( p ≥ s + 1 ) 的 s 阶隐式 Runge-Kutta 方法是矫顽且代数稳定的，那么我们可以证明所提出的方法是稳定的且收敛于 s 阶+ 1 及时。此外，我们还推导出了分布式阶项和空间项离散化的最优误差估计。最后，给出了数值实验来验证理论结果。