# Instabilities of the Forward Euler Method¶

Most of the PDE solvers in CC3D use Forward Euler explicit numerical scheme. This method is unstable for large diffusion constant. As a matter of fact using D=0.25 with pulse initial condition will lead to instabilities in 2D. To deal with this you would normally use implicit solvers however due to moving boundary conditions that we have to deal with in CC3D simulations, memory requirements, performance and the fact that most diffusion constants encountered in biology are quite low (unfortunately this is not for all chemicals e.g. oxygen ) we decided to use explicit scheme. If you have to use large diffusion constants with explicit solvers you need to do rescaling:

1. Set D, Δt, Δx according to your model
2. If
\begin{eqnarray} D\frac{\Delta t}{\Delta x^2}>0.16 {\text in 3D} \end{eqnarray}

you will need to to call solver multiple times per MCS.

1. Set <ExtraTimesPerMCS> to N-1 where:
\begin{eqnarray} ND' = D \end{eqnarray}

and

\begin{eqnarray} D\frac{\Delta t/N}{\Delta x^2} < 0.16 {\text in 3D} \end{eqnarray}

# Initial Conditions¶

We can specify initial concentration as a function of x, y, z coordinates using <InitialConcentrationExpression> tag we use muParser syntax to type the expression. Alternatively we may use ConcentrationFileName tag to specify a text file that contains values of concentration for every pixel:

<ConcentrationFileName>initialConcentration2D.txt</ConcentrationFileName>


The value of concentration of the last pixel read for a given cell becomes an overall value of concentration for a cell. That is if cell has, say 8 pixels, and you specify different concentration at every pixel, then cell concentration will be the last one read from the file.

Concentration file format is as follows:

*x y z c*


where x , y , z , denote coordinate of the pixel. c is the value of the concentration.

Example:

0 0 0 1.2

0 0 1 1.4

...


The initial concentration can also be input from the Python script (typically in the start function of the steppable) but often it is more convenient to type one line of the CC3DML script than few lines in Python.

# Boundary Conditions¶

All standard solvers (Flexible, Fast, and ReactionDiffusion) by default use the same boundary conditions as the GGH simulation (and those are specified in the Potts section of the CC3DML script). Users can, however, override those defaults and use customized boundary conditions for each field individually. Currently CompuCell3D supports the following boundary conditions for the diffusing fields: periodic, constant value (Dirichlet) and constant derivative (von Neumann). To specify custom boundary condition we include <BoundaryCondition> section inside <DiffusionField> tags.

The <BoundaryCondition> section describes boundary conditions along particular axes. For example:

<Plane Axis="X">
<ConstantValue PlanePosition="Min" Value="10.0"/>
<ConstantValue PlanePosition="Max"  Value="10.0"/>
</Plane>


specifies boundary conditions along the x axis. They are Dirichlet-type boundary conditions. PlanePosition='Min" denotes plane parallel to yz plane passing through x=0. Similarly PlanePosition="Min" denotes plane parallel to yz plane passing through x=fieldDimX-1 where fieldDimX is x dimension of the lattice.

By analogy we specify constant derivative boundary conditions:

<Plane Axis="Y">
<ConstantDerivative PlanePosition="Min" Value="10.0"/>
<ConstantDerivative PlanePosition="Max" Value="10.0"/>
</Plane>


We can also mix types of boundary conditions along single axis:

<Plane Axis="Y">
<ConstantDerivative PlanePosition="Min" Value="10.0"/>
<ConstantValue PlanePosition="Max" Value="0.0"/>
</Plane>


Here in the xz plane at y=0 we have von Neumann boundary conditions but at y=fieldFimY-1 we have dirichlet boundary condition.

To specify periodic boundary conditions along, say, x axis we use the following syntax:

<Plane Axis="X">
<Periodic/>
</Plane>


Notice, that <Periodic> boundary condition specification applies to both “ends” of the axis i.e. we cannot have periodic boundary conditions at x=0 and constant derivative at x=fieldDimX-1.