On the Convergence, Stability, and Computational Speed of Numerical Schemes for 0-D IC Engine Cylinder Modelling

摘要

The development of real time capable O-dimensional internal combustion engine models places high demands on convergence, stability, and computational speed of the applied numerical methods. The cylinder model represents the crucial element in attaining high computational speed and accuracy of results. A basic example comprising a single cylinder connected to two plenums is analysed with different numerical schemes in order to reveal methods effectively associating accuracy requirements with computational time constraints. The integration performance to solve a system of coupled ODEs was compared for explicit Euler and explicit fourth order Runge-Kutta schemes, as well as for multi-step methods including backward differentiation formulas and Adams-Moulton formulas. The performed analysis emphasizes two major points. First, the numerical accuracy of integration schemes differs significantly at equal computational effort revealing the necessity of selecting an adequate scheme for a specific task. Second, the comparison of integral engine parameters (e.g. indicated mean effective pressure, mean engine torque), calculated by different methods, with a numerically assumed exact solution should not be used as an estimate for the convergence and stability of the applied numerical approach, since good agreement in integral parameters does not imply good agreement in cycle resolved traces of thermodynamic variables. This paper provides clear guidelines for selecting the appropriate numerical integration methods with respect to the intended application. Analyses are also based on innovative test examples. Finally, a comparison of numerical and experimental in-cylinder pressure traces is shown for a series production engine confirming the applicability and accuracy of the cylinder model.