Using a template engine as a computer algebra tool

TL;DR

This paper introduces a Julia library that automates the generation of Runge-Kutta numerical schemes using template substitution, enhancing flexibility, reducing errors, and facilitating comparison of different methods for solving ODEs.

Contribution

The paper presents a novel template-based approach for automatically generating and testing various Runge-Kutta methods in Julia, streamlining implementation and comparison.

Findings

Automated code generation reduces implementation errors.

Template approach enables easy modification and comparison of schemes.

Generated codes successfully solve standard ODE problems.

Abstract

In research problems that involve the use of numerical methods for solving systems of ordinary differential equations (ODEs), it is often required to select the most efficient method for a particular problem. To solve a Cauchy problem for a system of ODEs, Runge-Kutta methods (explicit or implicit ones, with or without step-size control, etc.) are employed. In that case, it is required to search through many implementations of the numerical method and select coefficients or other parameters of its numerical scheme. This paper proposes a library and scripts for automated generation of routine functions in the Julia programming language for a set of numerical schemes of Runge-Kutta methods. For symbolic manipulations, we use a template substitution tool. The proposed approach to automated generation of program code allows us to use a single template for editing, instead of modifying each individual function to be compared. On the one hand, this provides universality in the implementation of a numerical scheme and, on the other hand, makes it possible to minimize the number of errors in the process of modifying the compared implementations of the numerical method. We consider Runge-Kutta methods without step-size control, embedded methods with step-size control, and Rosenbrock methods with step-size control. The program codes for the numerical schemes, which are generated automatically using the proposed library, are tested by numerical solution of several well-known problems.