Defect Sampling in Global Error Estimation for ODEs and Method-Of-Lines PDEs Using Adjoint Methods

The importance of good estimates of the defect in the numerical solution of initial value problem ordinary differential equations is considered in the context of global error estimation by using adjoint-equation based methods. In the case of solvers based on the fixed leading coefficient backward differentiation formulae, the quality of defect estimates is shown to play a major role in the reliability of the global error estimator of Cao and Petzold. New defect estimates obtained by sampling the defect are derived to improve the quality and efficiency of adjoint-based global error estimation. The inclusion of only one estimate of the defect per timestep is shown to provide a good compromise between accuracy and efficiency for global error estimation of odes and method-of-lines pdes. DEFECT SAMPLING IN GLOBAL ERROR ESTIMATION FOR ODES AND METHOD-OF-LINES PDES USING ADJOINT METHODS ∗ LETHUY TRAN† AND MARTIN BERZINS‡ Abstract. The importance of good estimates of the defect in the numerical solution of initial value problem ordinary differential equations is considered in the context of global error estimation by using adjoint-equation based methods. In the case of solvers based on the fixed leading coefficient backward differentiation formulae, the quality of defect estimates is shown to play a major role in the reliability of the global error estimator of Cao and Petzold. New defect estimates obtained by sampling the defect are derived to improve the quality and efficiency of adjoint-based global error estimation. The inclusion of only one estimate of the defect per timestep is shown to provide a good compromise between accuracy and efficiency for global error estimation of odes and method-of-lines pdes. The importance of good estimates of the defect in the numerical solution of initial value problem ordinary differential equations is considered in the context of global error estimation by using adjoint-equation based methods. In the case of solvers based on the fixed leading coefficient backward differentiation formulae, the quality of defect estimates is shown to play a major role in the reliability of the global error estimator of Cao and Petzold. New defect estimates obtained by sampling the defect are derived to improve the quality and efficiency of adjoint-based global error estimation. The inclusion of only one estimate of the defect per timestep is shown to provide a good compromise between accuracy and efficiency for global error estimation of odes and method-of-lines pdes.