Multi-Stage Time Stepping with Devito

This repository contains a Jupyter notebook implementing a multi-stage Runge-Kutta-like time integration scheme for solving PDE systems using Devito.

Overview

The notebook demonstrates a Multi-Stage Minimum Failure Example (MFE) Time Stepping Simulation that implements high-order time integration methods for partial differential equations.

Key Features

• Multi-stage Runge-Kutta time integration for PDE systems
• Source term handling with Taylor series expansion
• Strong Stability Preserving Runge-Kutta (SSPRK) coefficients
• Devito-based finite difference implementation
• 2D wave equation example with multiple source terms

Mathematical Foundation

The method formulates PDE systems as first-order in time:

∂U/∂t = HU + f(x,t)

where the solution is approximated using matrix exponentials and multi-stage Runge-Kutta methods following the approach from:

1. Al-Mohy AH, Higham NJ (2010) - Matrix exponential algorithms
2. Gottlieb S, Gottlieb LAJ (2003) - Strong stability preserving Runge-Kutta methods

Files

• MFE_time_size.ipynb - Main Jupyter notebook with complete implementation

Requirements

• Python 3.7+
• NumPy
• SymPy
• Devito

Installation

1. Install Devito following the official installation guide
2. Clone this repository:

   git clone https://github.com/fernanvr/time-stepping-devito.git
   cd time-stepping-devito

3. Launch Jupyter and open the notebook:

   jupyter notebook MFE_time_size.ipynb

Usage

The notebook is self-contained and includes:

1. Environmental setup - Grid and variable configuration
2. PDE system definition - Wave equation with source terms
3. Source derivatives computation - Symbolic differentiation
4. Operator construction - Multi-stage method implementation
5. Simulation execution - Running the Devito operator

Simply run all cells in sequence to execute the complete simulation.

Methodology

The implementation follows these key steps:

1. Formulate the PDE system as first-order in time
2. Expand source terms using Taylor series
3. Apply multi-stage Runge-Kutta integration
4. Use Devito for efficient finite difference computation

Results

The simulation produces time-evolved wavefields demonstrating the effectiveness of the multi-stage time integration approach for wave propagation problems.

License

This project is open source. Please cite the relevant papers when using this code in academic work.

Author

Fernando V. R. - GitHub

References

• Al-Mohy AH, Higham NJ (2010) A new scaling and squaring algorithm for the matrix exponential. SIAM Journal on Matrix Analysis and Applications 31(3):970–989
• Gottlieb S, Gottlieb LAJ (2003) Strong stability preserving properties of runge–kutta time discretization methods for linear constant coefficient operators. Journal of Scientific Computing 18(1):83–109