Modular, general purpose ODE integration package to solve large number of independent ODE systems on GPUs

TL;DR

This paper introduces a modular GPU-based package for efficiently solving large sets of independent ODE systems, supporting adaptive and fixed-step methods, with features for event detection and handling non-smooth dynamics.

Contribution

The paper presents a flexible, GPU-accelerated ODE integration package that enables efficient processing of many independent systems with customizable features for special solution properties.

Findings

Demonstrated efficiency with test cases like Duffing oscillator and Keller-Miksis equation.

Supports event detection and non-smooth dynamics handling.

Flexible pre-declared device functions for extracting solution features.

Abstract

A general purpose, modular program package for the integration of large number of independent ordinary differential equation systems capable of using professional graphics cards is presented. The available numerical schemes are the explicit and adaptive Runge--Kutta--Cash--Karp algorithm and the explicit fourth order Runge--Kutta method with fixed time step. In order to harness the huge processing power of graphics cards, the intermediate points of the computed trajectories are not stored. As a compensate, with pre-declared device functions, the required special features or properties of a solution can be easily extracted and stored each into a dedicated variable. For instance, the maximum and minimum values and/or their time instances. Event handling is also incorporated into the package in order to detect special points which can be stored as well. Moreover, again with pre-declared device function calls at such special points, the efficient handling of non-smooth dynamics---e.g. impact dynamics---is possible. Several test cases are presented to demonstrate the flexibility of the pre-declared device functions and the strength of the program package. The applied models are the simple Duffing oscillator, the more complex Keller--Miksis equation known in bubble dynamics, and a system describing the behaviour of a pressure relief valve that can exhibit impact dynamics.