Diffusion

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Simulations of a diffusing field

Introduction

Here, we build up a simulation of diffusion over three successive files by:

  1. Starting with 1D and a short time scale in Diffusion1a_main.xml ( | ) ,
  2. simulating for longer times in Diffusion1b.xml ( | ) and
  3. extending to 2D in Diffusion2a.xml ( | ) .

Description

Chemical diffusion and decay are easily simulated in Morpheus in 1D or 2D. We demonstrate the numerical simulation of the PDE and boundary conditions:

$$\begin{align} \frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2} - kc \quad c(0) = 0, c(L) = 5 \end{align}$$

Results

Building up a simulation of diffusion in Morpheus: (a) Starting with 1D and short time scale (Diffusion1a_main.xml ( | )), (b) for longer times (Diffusion1b.xml ( | )), and (c) extension to 2D (Diffusion2a.xml ( | )).

Model

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  •  Download: Diffusion1a_main.xml
  • XML Preview

    <?xml version='1.0' encoding='UTF-8'?>
    <MorpheusModel version="4">
        <Description>
            <Details>Full title:		Diffusion
    Authors:		L. Edelstein-Keshet
    Contributors:	Y. Xiao
    Date:		23.06.2022
    Software:		Morpheus (open-source). Download from https://morpheus.gitlab.io
    Model ID:		https://identifiers.org/morpheus/M2017
    File type:		Main model
    Reference:		L. Edelstein-Keshet: Mathematical Models in Cell Biology
    Comment:		A 1D domain with a chemical that diffuses at rate D and decays at rate k. Simple time plots in 1D of the chemical to show its diffusion into the domain.</Details>
            <Title>Diffusion1a</Title>
        </Description>
        <Space>
            <Lattice class="linear">
                <Neighborhood>
                    <Order>1</Order>
                </Neighborhood>
                <Size symbol="size" value="100, 0, 0"/>
                <BoundaryConditions>
                    <Condition type="constant" boundary="x"/>
                    <Condition type="constant" boundary="-x"/>
                </BoundaryConditions>
            </Lattice>
            <SpaceSymbol symbol="space"/>
        </Space>
        <Time>
            <StartTime value="0"/>
            <StopTime value="50"/>
            <TimeSymbol symbol="time"/>
        </Time>
        <Global>
            <Field symbol="c" value="0">
                <Diffusion rate="0.5"/>
                <BoundaryValue boundary="x" value="5.0"/>
            </Field>
            <System time-step="0.5" solver="Runge-Kutta [fixed, O(4)]">
                <DiffEqn symbol-ref="c" name="attractant">
                    <Expression>-k*c</Expression>
                </DiffEqn>
                <Constant symbol="k" name="decay rate" value="0.001"/>
            </System>
        </Global>
        <Analysis>
            <Logger time-step="5" name="chemical profile">
                <Input>
                    <Symbol symbol-ref="c"/>
                </Input>
                <Output>
                    <TextOutput/>
                </Output>
                <Plots>
                    <Plot>
                        <Style style="lines"/>
                        <Terminal terminal="png"/>
                        <X-axis>
                            <Symbol symbol-ref="space.x"/>
                        </X-axis>
                        <Y-axis>
                            <Symbol symbol-ref="c"/>
                        </Y-axis>
                    </Plot>
                </Plots>
            </Logger>
            <ModelGraph format="svg" reduced="false" include-tags="#untagged"/>
        </Analysis>
    </MorpheusModel>
    
    

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