基于非标准离散的非线性抛物型问题的时间外推算法

Nonlinear parabolic equations are widely used in practical scientific and engineering computing. Based on the virtual element method suitable for complicated regions and meshes, this project proposes to construct a new robust time extrapolation algorithm for such problems, including matrix time extrapolation algorithm and function time extrapolation algorithm, so as to reduce the formation of coefficient matrices and Jacobi matrices , and reduce the number of Newton iteration per time layer, thereby reducing the computing effort. To further improve the efficiency, the project will also develop the extrapolation cascadic multigrid method based on virtual element method and build the related mathematical theory. Finally, we combine these algorithms with the adaptive virtual element method, and apply them to the radiation diffusion equations with three-temperature to construct robust numerical methods with integrity, high efficiency and precision.

非线性抛物型方程在实际科学与工程计算中有广泛应用。基于适用于复杂区域和复杂网格的虚拟元方法，本项目拟构造这类问题新型的稳健时间外推算法，其包括矩阵时间外推方法和函数时间外推方法，以减少形成系数矩阵与Jacobi矩阵的次数以及每个时间层Newton迭代的次数，从而减少计算工作量。为进一步提高计算效率，本项目还将构造虚拟元方法的外推瀑布多网格方法，并建立其数学理论。最后，结合空间离散的自适应虚拟元方法，将这些算法推广应用于三温辐射扩散方程组，构建具有保正性、高效高精度稳健的数值方法。