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28 Feb 2012 (Updated )

A series of finite volume m-files for solving the linear advection and shallow water equations.

FVlinearadvectionFOUpseudo1D
function FVlinearadvectionFOUpseudo1D
% File name: FVlinearadvectionFOUpseudo1D.m
% Author: Clive Mingham
% Date: 23 Feb 2011
% Description:  Solves the PDE, dU/dt + div(H) = 0
% using the FOU finite volume scheme on a uniform Cartesian mesh inclined
% at theta degrees to the x-axis.  This is a pseudo-1D problem.  There is
% a single row of NI rectangular cells in the i direction with side lengths
% di and dj.  Flow velocity v has been fixed to ensure that there is no flow
% in the j direction.
%
% This equation corresponds to the finite volume form:
% doubleintegral(dU/dt dA) + lineintegral(H.n ds) = 0.
%
% In this case the equation is the linear advection equation so: 
% velocity: v = vx i + vy j where vx, vy are constant.
% flux density: H = vU = vx U i + vy U j.
%
% Cell areas and side vectors are the same for each cell.
%
% The program is written in a structured way using subfunctions so that it
% can be modified easily to pseudo-1D or 2D casees on non-Cartesian
% structured meshes.
%
% Initial conditions:  Gaussian along 1D line of cell centres.
%
% Boundary conditions:  Left: zero gradient.
%
% subfunctions: freadmesh, fcellarea, fsidevectors, finitialu,
%               finterfaceflux, fghostcells, fcalcdt, fplotresults, 
%               fdrawmesh
%
% Verification: Looks correct - numerical and exact solutions agree
% and the profile has travelled the correct distance.  Proper verification
% requires a pen and paper calculation on a small mesh.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
clc; clf; clear all;
runtime=10.0; % runtime in seconds
t=0;          % current time
timelevel=0;  % current time level
[x,y,xcen,ycen,solid,v]=freadmesh; % gets coordinates of the 1D computational mesh 
              % and solid flags (excluding ghost cells) and flow velocity
              % Note that there are no solid interior cells in 1D.
disp('mesh read in')
[m,n]=size(x); 
NI=m-1; % number of computational cells in i direction.
A=fcellarea(x,y);   % computes constant cell area
disp('calculated cell area')
S=fsidevectors(x,y); % compute and store cell side vectors                                         
disp('calculated cell side vectors')
u0=finitialu(xcen,ycen,t,v,S); % Put initial cell centre values of u in a NIx1 array
uinitial=u0; % store initial profile for plotting
disp('inserted initial conditions')
% Append extra cells around arrays to store any ghost values.
u0=finsertghostcells(u0);  % u0 is now (NI+2)x1
u1=zeros(size(u0));  % (NI+2)x1 array for u values at next time level
%%
disp('time marching starts')   
while(t<runtime)   
 timelevel=timelevel+1  
 u0=fbcs(u0);  % Implement boundary conditions using ghost cells. 
               % u0 is (NI+2)x1 and each computational cell is 
               % indexed i=2 to NI+1.
 dt=fcalcdt(A,S,v); % Finds stable time step for each iteration.
 t=t+dt  % update time  
   for i=2:NI+1 
       IH=finterfaceflux(v,u0,i); % gets the left and right interface fluxes for cell i
       IHright=[IH(1,1),IH(1,2)]; % interface flux vector right side 
       IHleft=[IH(2,1),IH(2,2)];  % interface flux vector for left side
       %
       sright=[S(1,1),S(1,2)]; % side vector for right side
       sleft=[S(2,1),S(2,2)];  % side vector for left side
       % FV scheme
       % compute total flux out of cell i
       totalfluxout=dot(IHright,sright)+dot(IHleft,sleft);
       
       % totalfluxout=vx*u0(i,1)*dy+vx*u0(i-1,1)*(-dy); WORKS!!!!!!!
       u1(i,1)=u0(i,1)-(dt/A)*totalfluxout; 
   end % of i loop
 %       
   u0=u1;  % update u values
end  % of while loop
uexact=finitialu(xcen,ycen,t,v,S); % find exact solution
t  % print time
subplot(2,1,1),fdrawmesh(x,y,solid)  
%fdisplay(u0); % print results for a small mesh as a matrix
subplot(2,1,2), fplotresults(xcen,ycen,u0,uinitial,uexact) % plot results
disp('program ended at time t see plots')
end % FVlinearadvectionFOUpseudo1D
%%%%%%%%%%%%%%%%%%%%%%%%%%% subfunctions  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [x,y,xcen,ycen,solid,v]=freadmesh
% Verification:  looks correct using fdrawmesh
% The mesh is structured and has NI cells in the i direction and 
% 1 cell in the j direction.
% The x and y coordinates of the lower left hand corner of cell (i,j) are
% held in arrays x and y respectively which are both (NI+1)x2.  In
% this way the 4 vertices of cell (i,j) are (x(i),y(j)), (x(i+1),y(j)),
% (x(i+1),y(j+1)) and (x(i),y(j+1)).  solid is an NI by 1 array which 
% flags solid cells.  If cell (i,j) is solid then solid(i,j)=1 otherwise
% solid(i,j)=0.
% x(1,1)=0=y(1,1).
% The mesh is created as a single row of uniform cells in the i direction
% which is a straight line inclined at theta degrees to the x axis.  
% This problem is pseudo-1D.
% v=[vx,vy]: vy component is chosen so that there is no flow in 
% the j direction.
% 
NI=100;  % number of cells in the i direction
x=zeros(NI+1,2); % allocate correct sized array for cell vertices
y=zeros(NI+1,2); % allocate correct sized array for cell vertices
xcen=zeros(NI,1); % allocate correct sized array for cell centres
ycen=zeros(NI,1); % allocate correct sized array for cell centres
solid=zeros(NI,1); % allocate correct sized array for solid cell flags
di=1.0; % cell length in i direction
dj=2.0; % cell length in j direction (arbitrary)
theta=pi/4;  % inclination angle (anticlockwise)
m=tan(theta); % slope of line
vx=sqrt(2);   % velocity component in x direction
vy=m*vx; % correct velocity component in y direction
% note that for theta=pi/4 the above ensure that flow is only in the
% i direction and the flow speed is 2 m/s.
v=[vx,vy]; % flow velocity vector.
dx=di*cos(theta);  % mesh increment in x direction
dy=di*sin(theta);  % mesh increment in y direction
x(1,1)=0; % first row starting point for x
y(1,1)=0; % first row starting point for x
x(1,2)=-dj*sin(theta); % second row starting point for x
y(1,2)=dj*cos(theta);  % second row starting point for y
% create uniform mesh
for i=2:NI+1
      x(i,1)=x(1,1)+(i-1)*dx;
      y(i,1)=y(1,1)+(i-1)*dy;
      x(i,2)=x(1,2)+(i-1)*dx;
      y(i,2)=y(1,2)+(i-1)*dy;
end % of i loop
%  x
%  y
% find cell centres by averaging coordinates of vertices 
% (this works for a general structured mesh)
for i=1:NI
     xcen(i,1)=(x(i,1)+x(i+1,1)+x(i+1,2)+x(i,2))/4;
     ycen(i,1)=(y(i,1)+y(i+1,1)+y(i+1,2)+y(i,2))/4;
end % of i loop
end % freadmesh
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
function A=fcellarea(x,y)  
% Verification: works but needs to be generalised for non-rectangular cells
% In a uniform Cartesian mesh cell area = di dj 
di=sqrt((x(2,1)-x(1,1))^2+(y(2,1)-y(1,1))^2);
dj=sqrt((x(1,2)-x(1,1))^2+(y(1,2)-y(1,1))^2);   
A=di*dj;  
end % fcellarea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
function S=fsidevectors(x,y)
% Verification: not verified
% For each cell: calculates right and left side vectors.
%
S=zeros(2,2);
%
rightside=[x(2,2)-x(2,1), y(2,2)-y(2,1)];
% Constant right side vectors
S(1,1)=rightside(2);
S(1,2)=-rightside(1);
% Constant left side vectors
leftside=[x(1,1)-x(1,2), y(1,1)-y(1,2)];
% Constant right side and unit normal vectors
S(2,1)=leftside(2);
S(2,2)=-leftside(1);
end % fsidevectors
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
function u=finitialu(xcen,ycen,t,v,S) 
% Verification:  verified
% Inserts u values at time t. 
[NI,NJ]=size(xcen);
u=zeros(NI,1);
rightside=[S(1,1),S(1,2)];
length=sqrt(dot(rightside,rightside)); % length of side vector
rightunitnormal=rightside/length;
speed=dot(v,rightunitnormal);  % flow speed in i direction
% Initial 1D Gaussian function of d where d is the distance along the line 
% connecting cell centres from (xcen(1,1),ycen(1,1)). Gaussian is based at d/2;
dmax=sqrt(((xcen(NI,1)-xcen(1,1))^2+(ycen(NI,1)-ycen(1,1))^2)); % max d value of domain
dc=dmax/3;  % approx mesh centre distance
%
cutoff=dmax/4 % cut off radius for Gaussian
for i=1:NI
        d=sqrt((xcen(i,1)-xcen(1,1))^2+(ycen(i,1)-ycen(1,1))^2)-speed*t;
        if (abs(d-dc)<cutoff)
           u(i,1)=exp(-0.01*(d-dc)^2);  % Gaussian
        else
           u(i,1)=0;
        end
end % of i loop
end % finitialu
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function IH=finterfaceflux(v,u0,i)
% Verification:  not verified
% Calculates right and left interface fluxes for each cell.
% Flux H depends on U, i.e. H = H(U). 
% For the 1D linear advection equation, H(U) = vx U i + 0 j.
% 
% There is considerable choice for interface flux estimation: different 
% choices give different FV schemes.
% 
%% The following FOU scheme simply takes:
%  IHright=H(u0(i,1)) = v u0(i,1)
%  IHleft=H(u0(i-1,1)) = v u0(i-1,1)
%  This works (is an upwind scheme) for vx > 0.
%
IH=zeros(2,2); % array to store left and right interface flux vectors
%
% Right
IH(1,1)=v(1)*u0(i,1);  % i component of right interface flux
IH(1,2)=v(2)*u0(i,1);  % j component of right interface flux
% Left
IH(2,1)=v(1)*u0(i-1,1);  % i component of left interface flux
IH(2,2)=v(2)*u0(i-1,1);  % j component of left interface flux
end % fIflux
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
function outarray=finsertghostcells(inarray)
% Verification:  not verified
% Assumes that ghost cells are needed at either end of the domain.
% array is embedded into (m+2)x1 array of zeros.  Hence computational 
% indices go from i=2 to m+1.  In the FOU scheme a ghost cell
% is needed only at the left end.
[m,n]=size(inarray);
dummy=zeros(m+2,1);
dummy(2:m+1,1)=inarray;
outarray=dummy;
end % finsertghostcells
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% function outarray=fremoveghostcells(inarray)
% Verification:  not verified
% % Removes ghost cells so that a mx1 array becomes (m-2)x1
% [m,n]=size(inarray);
% outarray=inarray(2:m-1,1);
% end % fremoveghostcells
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function u=fbcs(u)
% Verification:  verified
% Implements boundary conditions using ghost cells index by i=1, i=NI+2.
[m,n]=size(u);
NI=m-2;
% Left: zero gradient
u(1,1)=u(2,1);
% Right: not required for FOU scheme
end % bcs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function dt=fcalcdt(A,S,v)
% Verification:  not verified
% Finds allowable time step dt using heuristic formula.
% This formula must be generalised for a non Cartesian mesh.
rightside=[S(1,1),S(1,2)]; % right side vector
F=0.95; % tuning factor
dt=A/abs(dot(v,rightside)); % heuristic formula
dt=F*dt;
end  % fcalcdt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function fdrawmesh(x,y,solid)
% Verification:  not verified
% Description:  Draws a structured 2D finite volume mesh and fills in any
% solid cells. 
% Date structures:
% The mesh has NI cells in the i direction and NJ cells in the j direction.
% The x and y coordinates of the lower left hand corner of cell (i,j) are
% held in arrays x and y respectively which are both NI+1 by NJ+1.  In
% this way the 4 vertices of cell (i,j) are (x(i),y(j)), (x(i+1),y(j)),
% (x(i+1),y(j+1)) and (x(i),y(j+1)).  solid is an NI by NJ array which 
% flags solid cells.  If cell (i,j) is solid then solid(i,j)=1 otherwise
% solid(i,j)=0.
% 
[m,n]=size(x);
NI=m-1; % number of cells in i direction
NJ=1; % number of cells in j direction
%
% Plot the mesh
hold on   % keeps all plots on the same axes
% draw lines in the i direction
for i=1:NI+1
    plot(x(i,:),y(i,:),'k')
end
% draw lines in the j direction
for j=1:NJ+1
    plot(x(:,j),y(:,j),'k')   
end
title('computational mesh')
xlabel('x')
ylabel('y')
% Fill in solid cells
for i=1:NI
    for j=1:NJ
        if (solid(i,j)==1)
            solidx=[x(i,j),x(i+1,j),x(i+1,j+1),x(i,j+1),x(i,j)];
            solidy=[y(i,j),y(i+1,j),y(i+1,j+1),y(i,j+1),y(i,j)];
            fill(solidx,solidy,'k')      
        end
    end % of j loop
end % of i loop
end % fdrawmesh
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function fplotresults(xcen,ycen,u0,uinitial,uexact)
% Verification:  verified
% Dispays results along the mesh (i direction)
[NI,NJ]=size(xcen);
u0=u0(2:NI+1,1); % extract computational values
% find distances along the cell centre line
for i=1:NI
    d(i)=sqrt((xcen(i,1)-xcen(1,1))^2+(ycen(i,1)-ycen(1,1))^2);
end   
plot(d,uinitial,'k--',d,uexact,d,u0,'k.')
title('Graphs of initial profile (--) and numerical(.) and exact solutions')
xlabel('distance [m] along cell centre line of the pseudo-1D mesh')
ylabel('U [kg/m]')
end % fplotresults
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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