A Singly Diagonally Implicit Two-Step Peer Triple with Global Error Control for Stiff Ordinary Differential Equations

Abstract

This paper elaborates a new numerical method for treating stiff ordinary differential equations (ODEs) which constitute a basic simulation tool in many areas of study. For its efficiency, this technique is grounded in singly diagonally implicit two-step peer schemes designed recently. The latter means that linear systems with the same coefficient matrix are solved to advance a step of the method and to evaluate its global error as well. Moreover, in contrast to many other numerical methods with global error control, we monitor and regulate the scaled global error, which corresponds better with practitioners' needs. In other words, we present and study one peer triple which is able to solve stiff problems and to ensure that the true error of the derived numerical solution does not exceed the user-supplied bound at the same time. This implies that local and global error estimation and control mechanisms are also discussed here. Finally, efficiency of our numerical tool is checked in comparison to efficiency of stiff built-in MATLAB ODE solvers on various test problems.Key words. stiff ordinary differential equations, singly diagonally implicit peer methods, absolute and scaled local and global error estimations, automatic local and global error controls

Numerical robustness of extended Kalman filtering based state estimation in ill‐conditioned continuous‐discrete nonlinear stochastic chemical systems

“…It is well known that the local error control implemented in all MATLAB ODE solvers is not able to ensure the requested accuracy Tol of numerical integration in automatic mode and some sort of global error regulation is required for that. 61,[68][69][70][71] Therefore, using these two very different accuracy conditions in our MATLAB-based EKF methods is expected to gain confidence in conclusions drawn for our case studies in Section 4. Furthermore, we abbreviate these two ode45-based filtering techniques to EKF-ode45,10 −6 and EKF-ode45,10 −12 , respectively.…”

“…Thus, in order to regulate the committed discretization error and make it negligible in automatic mode, an ODE solver with automatic global error control is demanded for that. 61,[68][69][70][71] In this respect, a good choice is the embedded Gauss-type NIRK pairs of orders 4 and 6 with built-in local and global error control facilities developed in the work of Kulikov. 61 The cited ODE solvers have resulted in a number of accurate and efficient ACD-EKF methods published recently in related works.…”

Square-root filtering via covariance SVD factors in the accurate continuous-discrete extended-cubature Kalman filter

“…It is worth noting here that the continuous-discrete UKF methods with the NIRK pairs and built-in local and global error controls have been recently developed in conventional and Cholesky-based square-root forms in [26] and [31], respectively. Alternative advanced ODE solvers with automatic stepsize selection and error control facilities, especially those grounded in Runge-Kutta, general linear and peer methods published recently in [6,8,12,13,18,32,33,34,45,46,35,49,50,51,52,1], can contribute to nonlinear Bayesian filtering realm for treating complicated state estimation scenarios.…”