非线性抛物型偏微分方程奇异问题的高精度紧致有限体积法的开发及其数值分析

Many mathematical models describing natural and social phenomena with singularities are built in the form of nonlinear partial differential equations. How to get higher order numerical solutions of these partial differential equations is an urgent research subject in the field of numerical mathematics. In this project, we will study and develop higher order numerical methods for initial and boundary value problems of nonlinear parabolic partial differential equations with singular solutions. Since almost all the developed numerical methods are for the smooth problems, we will propose and develop compact finite volume methods with higher order accuracy. They are based on frameworks of finite element methods, finite volume methods and finite difference methods and thus are mixed higher order numerical methods which can be applied to the problems with singularities. The new compact finite volume methods developed for singular problems of nonlinear parabolic partial differential equations have the following significant advantages: (1) Higher order numerical approximate solutions are obtained without increasing computational costs and computational complexity; (2) The new numerical method is validate not only for singular points of the spatial domain, but also mainly for singularities arising on the whole boundaries or interfaces; (3) The refinement of meshes is adaptive, that is, the refinement is carried out automatically by the gradient of the exact solution; (4) The numerical approximate solution has super-convergence and the convergent order for singular problems is almost the same as for smooth problems; (5) The convergent order with respect to temporal discretization is of fourth order or higher order by integrating compact finite difference scheme with implicit methods; (6) Numerical analysis theory corresponding to the new compact finite volume methods is constructed so that error estimate analysis, super-convergence analysis and stability analysis can be carried out. The new compact finite volume methods will be widely applied to various practical fields such as chemical reaction, porous fluid dynamics, crystal and pattern formation, combustion and explosion and so on.

大量具有奇异性质的现象能够用偏微分方程数学模型描述。怎样获得这类方程的高精度数值解是计算数学领域中急需研究的课题。本项目将创新研究具有奇异性质的非线性抛物型偏微分方程的初边值问题的高精度数值方法。由于传统数值方法只适用于具有光滑性质的问题，本项目将提出并研究紧致有限体积法，它是基于解的梯度大小判别奇异性所出现的空间区域，利用细密剖分技术，将此部分加密生成网格，在其基础上构造有限体积法的基底函数，并综合有限元法处理边界条件的优势，有限体积法保持物理量的守恒性优势，来开发高精度紧致有限体积法。这种新的数值方法将不增加整体的计算量，又能近似达到和光滑问题的数值解同等程度的收敛阶数。另外，新的数值方法将具有自适应网格生成的特性。本项目也将建立对应的误差分析、稳定性分析等数值分析理论，以获得超收敛分析和稳定性分析结果。新的数值方法将在化学反应、多孔介质流体力学、燃烧爆炸等实际领域中获得广泛应用。

在自然科学领域以及社会科学领域中，非常多的线性现象和非线性现象都需要用常微分方程和偏微分方程建立数理模型进行表征。通过微分方程的解，不仅能够更加深刻地了解这些现象的存在原理，还能够预测这些现象的发生和发展。因此，求解微分方程，既具有数学理论上的重要意义，更具有科学技术实际上的应用价值。在许多以微分方程描述的数理模型问题中，含有各种各样的奇异性质。由于解决奇异性问题的难度远远超过研究光滑性问题的难度。特别当解的奇异性表现为解的导数发散于无穷大时，传统的有限差分法和有限元法以及相对应的数值分析理论就不再适用，需要建立新的数值方法和新的数值分析理论。本课题针对非线性抛物型偏微分方程奇异问题进行研究，针对解的奇异性所出现的空间区域，构造的部分区域分解法；分析在不具有奇异性的空间区域和具有奇异性的空间区域分别使用不同的网格细密剖分技术；利用有限元法处理边界条件的优势开发的高精度Compact Finite Volume Method；针对所开发的数值方法进行数值试验分析。针对全离散线性系统的特有性质，讨论建立奇异矩阵的迭代分析理论。本课题开发高精度紧致有限体积法。这种新的数值方法将不增加整体的计算量，又能近似达到和光滑问题的数值解同等程度 的收敛阶数。另外，新的数值方法将具有自适应网格生成的特性。本项目也将建立对应的误差分析、稳定性分析等数值分析理论，以获得超收敛分析和稳定性分析结果。新的数值方法将在化学反应、多孔介质流体力学、燃烧爆炸等实际领域中获得广泛应用。

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