Abstract

The error caused by inaccuracy in solving systems of equations by iterative methods is investigated. An upper estimate for an axially symmetric heat conduction equation is found for the error accumulated in several time steps. The estimate shows a linear dependence of the error on the threshold value of a criterion for limiting the number of iterations, a quadratic growth of the error depending on the number of points in space, and its independence of the number of steps in time. A computational experiment shows good agreement of the estimate with real errors at boundary and initial conditions of various types. A quadratic growth for Laplace’s equation of the error caused by an accuracy limitation in using an iterative method, depending on the number of points in space \(n\), is found empirically. A growth of \(n^{4}\) for the similar error in the biharmonic equation is found.

中文翻译：

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用迭代法求解数学物理简单方程的误差研究

摘要

研究了迭代法求解方程组不准确引起的误差。对于在几个时间步长中累积的误差，找到了轴对称热传导方程的上估计值。该估计显示误差与限制迭代次数的标准的阈值的线性相关性，误差的二次增长取决于空间中的点数，以及它与时间步数的独立性。计算实验表明，在各种类型的边界和初始条件下，估计值与实际误差具有良好的一致性。拉普拉斯方程的二次增长由使用迭代方法的精度限制引起的误差的二次增长，取决于空间\(n\) 中的点数，根据经验找到。的成长\(n^{4}\)发现了双调和方程中的类似误差。