【摘要】 常用的有限差分法、有限元方法和有限体积法等在数值求解偏微分方程时已经非常成功,但在处理各向异性问题时数值格式的保正性还存在一些问题.基于Feynman-Kac公式,可以将抛物方程的解表示为一个条件数学期望,涉及的随机扩散过程对应于抛物方程中的扩散项.不同于常用的紧致Markov链近似,本文用有限个连续分支路径逼近原来的随机过程,在满足相容性精度的条件下计算每个分支的停时(stopping time)、停时发生的概率和停时收益,从而可以逼近条件期望.这样基于数学期望的保正性,对任意的线性抛物型方程设计了一个全新的保正性的线性相容的数值格式,这是一个大时间步长、非紧致的、稳定的显式格式.由于有底层系统的理论支撑,本文的算法能够自适应地区分及处理边界的信息,避免现有算法(如半Lagrange方法)在边界附近精度缺失的问题.更多还原

【Abstract】 The finite difference method, finite element method, and finite volume method have achieved great success in the numerical solution of partial differential equations, but there are still some problems in designing the positivity-preserving scheme for anisotropic problems. Based on the Feynman-Kac formula, the solution of the parabolic differential equation can be formulated as a conditional mathematical expectation. The underlying stochastic diffusion process corresponds to the 2nd-order diffusion term of the parabolic equation. Unlike the common discrete compact Markov chain approximation, we approximate the original stochastic process by a continuous process with finite branches(paths), and calculate the stopping time(the time of hitting the boundary or reaching the next time step), the probability of stopping time and the profit of each branch under the condition to keep the sufficient consistency accuracy. Then we obtain the approximate conditional expectation or the numerical solution. In this way, for any linear parabolic equation, we design a new positivity-preserving, linear,and consistent numerical scheme, which is an explicit, stable, and noncompact one with a large time-step. Thanks to the solid theoretical support, our algorithm can adaptively handle the information of the boundary, and avoid the drawback that the semi-Lagrangian method loses its accuracy near the boundary.更多还原