Scaling parameters in central difference solution

[robby991]

Hi, I developed Matlab code to solve the diffusion equation using the central difference equation, with an added term at the end. The equation is the following:

 dS/dt=Ds*d^2S/dx^2-(Vmax*S/Km+S)

In my code, the length of the space domain is very small, 1E-6. I would like to scale my code to run in the space domain between x = 0 and 1, rather than x = 0 to 1E-6. I propose to introduce a new variable, x2 = x/L. My question is what parameters/variables in my code need to be scaled also. My code is:

clear all;

numx = 10;                      %number of grid points in space
numt = 1000;                    %number of time steps to be iterated over 
tmax = .0045;
Length = 1E-6;                  %length of grid
Ds = .019E-9;                   %requirement Ds(dt)/dx^2 < .5, cm^2/sec
Vmax = 275E-6;                  %mol/cm^2sec
Km = 3E-3;                      %mol/cm^3

x = linspace(0,Length,numx);    %vector of x values, to be used for plotting
t = linspace(0,tmax,numt)';     %vector of t values, to be used for plotting
S = zeros(numt,numx);           %initialize everything to zero

dx = x(2)-x(1);                 %Define grid spacing in time
dt = t(2)-t(1);                 %Define grid spacing in time

%specify initial conditions%

t(1) = 0;      %1st t position = 0

S(1,:) = cos(x*pi/(2*Length));   
S(:,numx) = 0;

S_exact = cos(x*pi/(2*Length))*exp(-(pi*sqrt(Ds)/(2*Length))^2*tmax);

%iterate central difference equation% 

for j=1:numt-1  
    
      
    %2nd Derivative Central Difference Iteration% 
     
    for i=2:numx-1
      S(j+1,i) = S(j,i) + (dt/dx^2)*Ds*(S(j,i+1) - 2*S(j,i) + S(j,i-1))-((Vmax*dt*S(j,i))/(Km+S(j,i))); 
    end
    
    S(j+1,1)=S(j,1)+dt*Ds*2*(S(j,2)-S(j,1))./dx.^2-((Vmax*dt*S(j,i))/(Km+S(j,i)));      %Neumann Boundary Condition
    
end
   
plot(x,S(numt,:));  
hold on
plot(x,S_exact,'r*')

error = max(S(numt,:)-S_exact)

To scale, the domain would now be:

x2=x/Length;

and now, dx = x2(2)-x2(1);

How do I change me code to accommodate this? How do my other variables change/scale? i.e. Km, Ds, Vmax etc. Thank you.

 

[AlephZero]

This is exactly the same as working in different units, for example changing your lengths from km to mm (as an example of units which happen to be a factor of 10^6 different)

If you change the values of ALL the constants in your code which have physical dimensions, you don't need to change anything else.

 

[robby991]

I have one last question regarding this. Do I have to scale the matrix vectors in the time domain also? I scaled all the parameters related to the space domain (x), but I also have the time vectors (y). If so, do I divide by the same constant I used to scale the space domain?

 

[robby991]

I ran into a little difficulty and was wondering if my scaling of parameters was correct in my code. Say I have a grid of length "Length", and instead of going from 0 to Length I want to scale from 0 to 1. In addition I have the following constants in my calculations:

Length = 1E-4;                  %cm
Ds = .019E-9;                   % cm^2/sec
Vmax = 275E-6;                  %mol/cm^2sec
Km = 3E-3;                      %mol/cm^3

To scale from 0 to 1, I would simply make a new variable L where:

L = Length/Length.

Next, to scale the other variables I will do the following:

D = Ds/Length^2
V = Vmax*Length^2
K = Km*Length^3

Is this correct? I would appreciate any input on the matter. Thank you.