积分延迟修正（IDC）方法已被证明是实现任意高阶精度的有效方式，并具有良好的稳定性特性。本文利用IDC过程构建高阶算子分裂方案，用于求解初始值问题（IVPs）。我们分析表明，IDC方法能够同时修正分裂误差和数值误差，通过每次修正提升精度阶数$ r $，其中$ r $是用于求解修正方程的方法的精度阶数。我们进一步将此框架应用于求解偏微分方程（PDEs）。通过二维线性和非线性初边值问题的数值例子，展示了所提出的IDC方法的性能。
英文摘要：
Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the IDC procedure to solve initial value problems (IVPs). We present analysis to show that the IDC methods can correct for both the splitting and numerical errors, lifting the order of accuracy by $r$ with each correction, where $r$ is the order of accuracy of the method used to solve the correction equation. We further apply this framework to solve partial differential equations (PDEs). Numerical examples in two dimensions of linear and nonlinear initial-boundary value problems are presented to demonstrate the performance of the proposed IDC approach.
The paper investigates a variant of semi-implicit spectral deferred corrections (SISDC) in which the stiff, fast dynamics correspond to fast propagating waves ("fast-wave slow-wave problem"). We show that for a scalar test problem with two imaginary eigenvalues $i λ_{fast}$, $i λ_{slow}$, having $Δt \left(\left| λ_{fast} \right| + \left| λ_{slow} \right| \right) < 1$ is sufficient for the fast-wave slow-wave SDC (FWSW-SDC) iteration to converge and that in the limit of infinitely fast waves the convergence rate of the non-split version is retained. Stability function and discrete dispersion re…
This paper presents a semi-implicit spectral deferred correction (SDC) method for incompressible Navier-Stokes problems with variable viscosity and time-dependent boundary conditions. The proposed method integrates elements of velocity- and pressure-correction schemes, which yields a simpler pressure handling and a smaller splitting error than the SDPC method of Minion & Saye (J. Comput. Phys. 375: 797-822, 2018). Combined with the discontinuous Galerkin spectral-element method for spatial discretization it can in theory reach arbitrary order of accuracy in time and space. Numerical experiment…
Parallel performance of shared memory parallel spectral deferred corrections
Hamburg University of Technology（汉堡工业大学）；Univ. Grenoble Alpes / Laboratoire Jean Kuntzmann / Inria（格勒诺布尔阿尔卑斯大学/让·孔特实验室/法国国家信息与自动化研究所）；Technical University of Munich（慕尼黑工业大学）
We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components which describe the coupling and the dynamics in the bulk and on the surface, and treats the nonlinear terms by approximating the integral in the variation of constants formula. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups and a careful comparison of the exact and approximate flows. Numerical experiments, including problems with dynami…
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A framework is developed for a robust and highly accurate numerical solution of the coupled Stokes-Darcy system in three dimensions. The domain decomposition method is based on a Dirichlet-Neumann type splitting of the interface conditions and solving separate Stokes and Darcy problems iteratively. Second kind boundary integral equations are formulated for each problem. The integral equations use a smoothing of the kernels that achieves high accuracy on the boundary, and a straightforward quadrature to discretize the integrals. Numerical results demonstrate the convergence, accuracy, and depen…
We propose a short-memory operator splitting scheme for solving the constant-Q wave equation, where the fractional stress-strain relation contains multiple Caputo fractional derivatives with order much smaller than 1. The key is to exploit its extension problem by converting the flat singular kernels into strongly localized ones, so that the major contribution of weakly singular integrals over a semi-infinite interval can be captured by a few Laguerre functions with proper asymptotic behavior. Despite its success in reducing both memory requirement and arithmetic complexity, we show that numer…
