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

Abstract

Solving initial value problems (IVPs) for linear ordinary differential equations (ODEs) plays an important role in many applications. There are variousnumerical methods and solvers to obtain approximate solutions typically represented by points. However, few works about estimation of global errors canbe 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.