Integration Algorithms used in Simcenter Amesim

Introduction

This technical note describes: the algorithms Simcenter Amesim uses to solve/integrate the equations defining a model, the reasons for that choice, and the strategy for progressively upgrading these algorithms.

When developing models, assumptions and simplifications are inherent — deciding what these should be is a modeler's skill. Very fine-detail models would treat important variables as functions of both time and position (e.g. pressure P as a function of 4 variables — 3 spatial + time), requiring a large grid of nodes and arrays of state variables governed by systems of partial differential equations (PDEs). Specialist packages (e.g. CFD software) solve equations of this type for detailed analysis of individual components, but Amesim's system-level models instead work with lumped-parameter ODEs.

The Range of Methods Available

A numerical solution isn't obtained continuously over 0 <= t <= tfin; instead it's obtained at a finite collection of discrete times, with the approximate solution for y at t = tn denoted yn.

All widely used ODE integration methods fall into two categories:

• Explicit Runge-Kutta (ERK) methods
• Linear multistep methods — referred to as "Simcenter Amesim methods" in this context

Terminology: a method is a particular formula used to integrate ODEs. An algorithm is the implementation of that formula together with step-size control and possible switching between different formulae.

The Algorithms used by Simcenter Amesim

The main objective when choosing Amesim's integration algorithms was to free the user from having to select an algorithm manually — even an experienced numerical mathematician finds this choice difficult, made worse because the characteristics of the governing equations can change as the simulation progresses. It's unreasonable to expect a user to notice this mid-run, stop the simulation, change algorithm, and restart. Instead, Amesim uses algorithms that automate these method-switching decisions reliably.