Abstract:
The problem of efficient global error estimation and control is studied in embedded nested implicit Runge–Kutta pairs of Gauss and Lobatto types as applied to stiff ordinary differential equations (ODEs). Stiff problems may arise in many areas of engineering, and their accurate numerical solution is an important issue of computational and applied mathematics. A cheap global error estimation technique designed recently for the mentioned Runge–Kutta pairs can severely overestimate the global error when applied to stiff ODEs and, hence, this reduces the efficiency of those solvers. In the present paper, we explain the cause of that error overestimation and show how to improve the mentioned computation techniques for stiff problems. Such modifications not only boost the efficiency of numerical integration of stiff ODEs, but also make the embedded nested implicit Runge–Kutta pairs with scaled modified local and global error controls superior to stiff built-in MATLAB ODE solvers with only local error control when applied to important test examples.
Key words:ordinary differential equation, stiff problem, embedded nested implicit Runge–Kutta pairs of Gauss and Lobatto types, absolute and scaled local and global error estimates, automatic local and global error controls.
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