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首页> 外文期刊>Journal of Computational Physics >High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements
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High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements

机译：普通光滑域的高阶无条件稳定FC-AD求解器I.基本元素

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摘要

We introduce a new methodology for the numerical solution of Partial Differential Equations in general spatial domains: our algorithms are based on the use of the well-known Alternating Direction Implicit (ADI) approach in conjunction with a certain "Fourier continuation" (FC) method for the resolution of the Gibbs phenomenon. Unlike previous alternating direction methods of order higher than one, which can only deliver unconditional stability for rectangular domains, the present high-order algorithms possess the desirable property of unconditional stability for general domains; the computational time required by our algorithms to advance a solution by one time-step, in turn, grows in an essentially linear manner with the number of spatial discretization points used. In this paper we demonstrate the FC-AD methodology through a variety of examples concerning the Heat and Laplace Equations in two and three-dimensional domains with smooth boundaries. Applications of the FC-AD methodology to Hyperbolic PDEs together with a theoretical discussion of the method will be put forth in a subsequent contribution. The numerical examples presented in this text demonstrate the unconditional stability and high-order convergence of the proposed algorithms, as well the very significant improvements they can provide (in one of our examples we demonstrate a one thousand improvement factor) over the computing times required by some of the most efficient alternative general-domain solvers.
机译：我们为一般空间域中的偏微分方程的数值解引入了一种新的方法：我们的算法基于著名的交替方向隐式（ADI）方法和某种“傅里叶连续”（FC）方法的结合使用解决吉布斯现象。不同于先前的只能提供矩形域无条件稳定性的高阶交替方向方法，本发明的高阶算法具有一般域无条件稳定性的理想特性。反过来，我们的算法将解决方案向前推进一个时间步所需的计算时间随着所使用的空间离散点的数量以基本上线性的方式增长。在本文中，我们通过具有光滑边界的二维和三维域中涉及热和拉普拉斯方程的各种示例，论证了FC-AD方法。 FC-AD方法在双曲PDE上的应用以及对该方法的理论讨论将在随后的文章中提出。本文中提供的数值示例证明了所提出算法的无条件稳定性和高阶收敛性，以及它们在计算所需的计算时间上可以提供的非常显着的改进（在我们的一个示例中，我们演示了一千个改进因子）一些最有效的替代通用域求解器。

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