论文标题
由卷积运算符的可变步骤近似产生的实际二次形式的正确定性
Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators
论文作者
论文摘要
带有卷积结构的实际二次形式的积极确定性在非本地运算符的时间步长方案中起着重要作用。在这项工作中，我们提供了一种新的分析工具，以处理由卷积运算符的可变步骤近似而导致的离散卷积内核。更确切地说，对于一类与可变时间离散相关的离散卷积内核，我们表明，在一些易于检查的代数条件下，相关的二次形式是积极的。我们的证明是基于使用离散正交卷积内核和互补卷积内核的基本构造策略。据我们所知，这是简单的代数条件下的第一个总体结果，它是可变离散卷积内核的积极确定性的。使用统一的理论，可以直接获得一些简单的不均匀时间稳定方案的稳定性。
The positive definiteness of real quadratic forms with convolution structures plays an important role in stability analysis for time-stepping schemes for nonlocal operators.In this work, we present a novel analysis tool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators. More precisely, for a class of discrete convolution kernels relevant to variable-step time discretizations,we show that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Our proof is based on an elementary constructing strategy using the properties of discrete orthogonal convolution kernels and complementary convolution kernels. To the best of our knowledge, this is the first general result on simple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using the unified theory, the stability for some simple non-uniform time-stepping schemes can be obtained in a straightforward way.
