Getting started with solvers in Engee

Fixed step - The step remains unchanged throughout the computational experiment. Usually used for real-time calculations or for checking the correctness of the numerical solution. Generally inefficient because the step over the entire simulation interval is limited by the requirements of certain more dynamic sections. Reducing the step size to a certain limit allows to increase the accuracy of the solution, but naturally has a negative effect on the speed of calculation;

Variable step - automatically adjust the step size as the computational experiment progresses, taking into account changes in the solution and estimates of local integration errors. Such solvers reduce the step in complex domains to maintain accuracy and increase it in simple domains to speed up the computation.

* Implicit methods* - require solving a system of nonlinear algebraic equations at each step. They are suitable for stiff systems (see Sect. Solver selection) because they allow large steps without loss of stability, but each step requires serious computational resources. Examples in Engee: ImplicitEuler, Trapezoid, TRBDF2, RadauIIA5, KenCarp*, Kvaerno*, QNDF*, QBDF*, FBDF;

Semi-implicit methods - unlike implicit methods require solving already a system of linear algebraic equations at each step. Like implicit ones, they are well suited for rigid systems of equations. In general, strongly depend on the accuracy of the calculation of the Jacobi matrix. Examples in Engee: Rosenbrock*, Rodas*;

Multistep methods - use data from several previous steps to compute the next one. They can be particularly effective for large problems, but in the presence of multiple events their speed drops due to constant reinitialisations, so they are not recommended for models with discontinuous components. Examples in Engee: VCABM*, AB*, ABM*, QNDF*, QBDF*, FBDF.

System rigidity - a rigid model is one in which fast and slow processes occur simultaneously, such as in chemical reactions or electromechanical systems. For such models explicit methods due to their small stability region require very small steps, so it is better to use implicit or semi-implicit methods, the step of which will be limited only by accuracy requirements. A system is considered rigid if its time constants differ by 6 or more orders of magnitude;

Model size - the number of continuous variables uniquely identifying a particular state of the model. A model is considered large if it has more than 1000 state variables. The more there are, the slower the computation, especially in usage of implicit and semi-implicit methods. For large models, it is better to use methods with fewer calculations of the right-hand side (derivatives of the system) at each step;

Requirements to the solution accuracy - determine the closeness of numerical and exact solutions. The requirements for absolute and relative accuracy are separately distinguished:

Gaps in the solution - some library components, e.g. Abs or Saturation, are characterised by sudden jumps in behaviour, resulting in gaps in the solution. Multistep methods are generally less efficient for such problems.