线性常微分方程的全局误差估计和优化求解方法

线性常微分方程初值问题求解在许多应用中起着重要作用.目前,已存在很多的数值方法和求解器用于计算离散网格点上的近似解,但很少有对全局误差(global error)进行估计和优化的方法.本文首先通过将离散数值解插值成为可微函数用来定义方程的残差;再给出残差与近似解的关系定理并推导出全局误差的上界;然后以最小化残差的二范数为目标将方程求解问题转化为优化求解问题;最后通过分析导出矩阵的结构,提出利用共轭梯度法对其进行求解.之后将该方法应用于滤波电路和汽车悬架系统等实际问题.实验分析表明,本文估计方法对线性常微分方程的初值问题的全局误差具有比较好的估计效果,优化求解方法能够在不增加网格点的情形下求解出线性常微分方程在插值解空间中的全局最优解.

Global error estimation for linear ordinary differential equations and their numerical optimal solutions

Solving initial value problems(IVPs)for linear ordinary differential equations(ODEs)plays an important role in many applications.There are various numerical methods and solvers to obtain approximate solutions typically represented by points.However,few works about estimation of global errors can be found in literature.In this paper,we first use Hermite cubic spline interpolation at mesh points to represent the solution,and then we define this residual obtained by substituting the interpolation solution back to ODEs.Then the global error between the exact solution and an approximate solution can be bounded by using the residual.Moreover,solving ODEs can be reduced to an optimization problem of the residual in certain solution space which can be solved by the conjugate gradient method by taking advantage of sparsity of the corresponding matrix.The examples in the paper show that our estimation works well for linear ODE models and the refinement can find solutions with smaller global errors than some popular methods in MATLAB without additional mesh points.