IMEX Solvers for Reaction–Diffusion PDEs

This repository collects a set of IMEX (implicit–explicit) time-stepping schemes developed for solving reaction–diffusion PDEs, and organises them into a reusable and extensible structure.

IMEX methods are designed for problems where different terms have different numerical properties. A general PDE can be written as:

$$ \frac{du}{dt} = \mathcal{L}(u) + \mathcal{N}(u) $$

where:

• $\mathcal{L}(u)$ is typically a stiff linear operator (e.g. diffusion)
• $\mathcal{N}(u)$ is a nonlinear term (e.g. reactions)

IMEX schemes treat these separately:

$$ \frac{u^{n+1} - u^n}{\Delta t} = \mathcal{L}(u^{n+1}) + \mathcal{N}(u^n) $$

so that:

• the stiff part is handled implicitly (stable)
• the nonlinear part is handled explicitly (cheap to compute)

This makes them particularly useful for PDEs where fully explicit methods are unstable and fully implicit methods are too expensive.

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Why this exists

During coursework, these schemes were implemented multiple times in separate notebooks and scripts. This repository brings them together into a single structure so they can be reused directly for other PDE models without rewriting the numerical method each time.

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What it does

The solvers are built for two-species systems of the form:

$$ \begin{aligned} u_t &= \nabla \cdot (D_u \nabla u) + \gamma f(u, v) \\ v_t &= \nabla \cdot (D_v \nabla v) + \gamma g(u, v) \end{aligned} $$

and also support extensions to nonlinear and variable diffusion, including terms of the form:

$$ \nabla \cdot (D(u, v) \nabla u) $$

This includes both standard diffusion and more general cases such as cross or density-dependent diffusion.

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Implemented methods

The following IMEX schemes are included.

AB1–AM1 (IMEX Euler)

$$ (I - \Delta t L) u^{n+1} = u^n + \Delta t , \gamma f(u^n) $$

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AB2–AM2

$$ (I - \tfrac{\Delta t}{2} L) u^{n+1} = (I + \tfrac{\Delta t}{2} L) u^n

• \Delta t \left( \tfrac{3}{2} f(u^n) - \tfrac{1}{2} f(u^{n-1}) \right) $$

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AB2–BDF2

$$ \left( \tfrac{3}{2} I - \Delta t L \right) u^{n+1} = 2 u^n - \tfrac{1}{2} u^{n-1}

• \Delta t \gamma \left( 2 f(u^n) - f(u^{n-1}) \right) $$

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AB4–BDF4

$$ (25 I - 12 \Delta t L) u^{n+1} = 48 u^n - 36 u^{n-1} + 16 u^{n-2} - 3 u^{n-3} $$

$$

• 12 \Delta t \gamma \left( 4 f(u^n) - 6 f(u^{n-1}) + 4 f(u^{n-2}) - f(u^{n-3}) \right) $$

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Structure

src/
├── solvers/
│   ├── ab1am1.py
│   ├── ab2am2.py
│   ├── ab2bdf2.py
│   └── ab4bdf4.py
│
├── spatial/
│   ├── laplacian_1d.py
│   ├── laplacian_2d.py
│   ├── variable_diffusion_1d.py
│   └── variable_diffusion_2d.py
│
├── models/
│   ├── schnakenberg.py
│   ├── gierer_meinhardt.py
│   └── tumor_ecm.py

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What’s next

• include additional IMEX schemes as they are implemented
• extend support for more general nonlinear diffusion operators
• apply these solvers to new PDE models

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License

This project is released under the MIT License. See the LICENSE file for details.

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Overall, this repository is intended as a reusable base for working with IMEX methods in PDEs, rather than a collection of one-off scripts.