Convergence properties of a class of boundary element approximations to linear diffusion problems with localized nonlinear reactions
Convergence properties of a class of boundary element approximations to linear diffusion problems with localized nonlinear reactions
复制标题
DOI：
10.1002/num.1690060106
复制
发表时间：
1990-03
影响因子：
3.9
通讯作者：
A. Peirce;A. Askar;H. Rabitz
中科院分区：
文献类型：
--
作者：
A. Peirce;A. Askar;H. Rabitz
我们考虑使用边界元 (BE) 算法来解决具有局部非线性反应的线性扩散解吸问题。所提出的 BE 算法提供了局部非线性反应效应的优雅表示，使得任意方向的缺陷结构的效应能够合并到 BE 模型中，而无需执行严重的网格变形。我们提出了一种一步递归程序来推进线性扩散局域非线性反应问题的 BE 解并研究其收敛特性。通过边界积分公式分离线性和非线性效应使我们能够分别考虑边界积分方程的线性项和非线性项的近似的收敛性质。对于线性项，我们研究了空间中分段多项式配置的程度以及空间网格相对于时间步长的大小如何影响一步递归方案中的误差累积。我们开发了一种新颖的收敛分析，将渐近方法与 Lax 等价定理相结合。我们确定了一个无量纲网格划分参数 θ，其大小决定了一步 BE 方案的性能。特别是，我们表明分段常数（PWC）和分段线性（PWL）BE方案是条件收敛的，在时间步长的大小上具有较低的渐近界限，并且当使用小时间步长时显示出过度的数值扩散。连接步长的大小没有渐近界限——这使得解决方案能够以更少、更大的时间步长推进。分段二次 (PWQ) BE 方案被证明是无条件收敛的；时间和空间网格的相对大小没有渐近限制，也没有数值扩散。我们在数值例子中验证了理论收敛性。该分析提供了有关空间分段多项式的适当程度和给定问题的网格划分策略的有用信息。对于非线性项，我们研究了显式算法的收敛性，通过 Caratheodory 迭代与分段线性插值相结合的方式及时推进活动站点处的解。我们考虑一个由奇异非线性 Volterra 方程组成的模型问题，该方程表示 BE 公式中的项由于单个缺陷而产生的影响。我们证明分段线性 Caratheodory 迭代算法对于模型问题的解的收敛性，只要可以证明这样的解存在。该分析为使用分段线性 Caratheodory 迭代来提高局部反应的效果提供了理论依据。
We consider a boundary element (BE) Algorithm for solving linear diffusion desorption problems with localized nonlinear reactions. The proposed BE algorithm provides an elegant representation of the effect of localized nonlinear reactions, which enables the effects of arbitrarily oriented defect structures to be incorporated into BE models without having to perform severe mesh deformations. We propose a one-step recursion procedure to advance the BE solution of linear diffusion localized nonlinear reaction problems and investigate its convergence properties. The separation of the linear and nonlinear effects by the boundary integral formulation enables us to consider the convergence properties of approximations to the linear terms and nonlinear terms of the boundary integral equation separately. For the linear terms we investigate how the degree of piecewise polynomial collocation in space and the size of the spatial mesh relative to the time step affects the accumulation of errors in the one-step recursion scheme. We develop a novel convergence analysis that combines asymptotic methods with Lax's Equivalence Theorem. We identify a dimensionless meshing parameter θ whose magnitude governs the performance of the one-step BE schemes. In particular, we show that piecewise constant (PWC) and piecewise linear (PWL) BE schemes are conditionally convergent, have lower asymptotic bounds placed on the size of time steps, and which display excess numerical diffusion when small time steps are used. There is no asymptotic bound on how large the tie steps can be–this allows the solution to be advanced in fewer, larger time steps. The piecewise quadratic (PWQ) BE scheme is shown to be unconditionally convergent; there is no asymptotic restriction on the relative sizes of the time and spatial meshing and no numerical diffusion. We verify the theoretical convergence properties in numerical examples. This analysis provides useful information about the appropriate degree of spatial piecewise polynomial and the meshing strategy for a given problem. For the nonlinear terms we investigate the convergence of an explicit algorithm to advance the solution at an active site forward in time by means of Caratheodory iteration combined with piecewise linear interpolation. We consider a model problem comprising a singular nonlinear Volterra equation that represents the effect of the term in the BE formulation that is due to a single defect. We prove the convergence of the piecewise linear Caratheodory iteration algorithm to a solution of the model problem for as long as such a solution can be shown to exist. This analysis provides a theoretical justification for the use of piecewise linear Caratheodory iterates for advancing the effects of localized reactions.