论文标题
高度振荡的数值解决方案的致密输出
Dense output for highly oscillatory numerical solutions
论文作者
论文摘要
我们提出了一种在数值解决方案的特殊情况下构建连续扩展(也称为密度输出)的方法,即具有标量值的函数,表现出快速振荡。这种情况要求使用解决方案的已知全局行为的数值例程,一个示例是使用渐近扩展的方法来预测自变量的每个步骤的解决方案。一个示例是OSCODE,数值例程,当解决方案迅速振荡并以其他方式表现为runge-kutta(rk)求解器时,它使用wentzel-kramer-brillouin(wkb)近似。多项式插值不适用于在任意点中间的任意点产生溶液,因为基于WKB近似的有效数值方法将在单个步骤中逐步跨越多个振荡。取而代之的是,我们通过扩展用于计算解决方案的WKB近似的数值正交构建连续解决方案,而没有对微分方程或术语内部的其他评估,并对此致密输出提供了错误估计。最后,我们将注意力集中在runge-kutta公式的连续扩展方面的先前工作,并基于Gauss--lobatto正交节点构建了RK方法的扩展,从而描述了如何从基础oscode的每种方法中产生密集输出。
We present a method to construct a continuous extension (otherwise known as dense output) for a numerical routine in the special case of the numerical solution being a scalar-valued function exhibiting rapid oscillations. Such cases call for numerical routines that make use of the known global behaviour of the solution, one example being methods using asymptotic expansions to forecast the solution at each step of the independent variable. An example is oscode, numerical routine which uses the Wentzel-Kramers-Brillouin (WKB) approximation when the solution oscillates rapidly and otherwise behaves as a Runge-Kutta (RK) solver. Polynomial interpolation is not suitable for producing the solution at an arbitrary point mid-step, since efficient numerical methods based on the WKB approximation will step through multiple oscillations in a single step. Instead we construct the continuous solution by extending the numerical quadrature used in computing a WKB approximation of the solution with no additional evaluations of the differential equation or terms within, and provide an error estimate on this dense output. Finally, we draw attention to previous work on the continuous extension of Runge-Kutta formulae, and construct an extension to a RK method based on Gauss--Lobatto quadrature nodes, thus describing how to generate dense output from each of the methods underlying oscode.