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Method of characteristics for two-dimensional advection equation

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Asatur Khurshudyan
Asatur Khurshudyan on 14 Apr 2016
Given the following function for implementing the method of characteristics:
function qnew = SemiLagrAdvect(u,v,q,qS,qN,qW,qE)
global N M
global dx dy
global dt Re
u = reshape(u,N,M);
v = reshape(v,N,M);
q = reshape(q,N,M);
%...embedding
qq = zeros(N+2,M+2);
qq(2:N+1,2:M+1) = q;
%...set the ghost values (four edges)
qq(1,2:M+1) = 2*qW-qq(2,2:M+1);
qq(N+2,2:M+1) = 2*qE-qq(N+1,2:M+1);
qq(2:N+1,1) = 2*qS-qq(2:N+1,2);
qq(2:N+1,M+2) = 2*qN-qq(2:N+1,M+1);
%...set the ghost values (four corners)
qq(1,1) = -qq(2,2);
qq(N+2,1) = -qq(N+1,2);
qq(N+2,M+2) = -qq(N+1,M+1);
qq(1,M+2) = -qq(2,M+1);
q1 = qq(2:N+1,2:M+1);
q2p = qq(3:N+2,2:M+1);
q2m = qq(1:N,2:M+1);
q3p = qq(2:N+1,3:M+2);
q3m = qq(2:N+1,1:M);
q4pp = qq(3:N+2,3:M+2);
q4mm = qq(1:N,1:M);
q4pm = qq(3:N+2,1:M);
q4mp = qq(1:N,3:M+2);
xi = -u*dt/dx;
eta = -v*dt/dy;
Q2 = q2p.*(xi>0) + q2m.*(xi<0);
Q3 = q3p.*(eta>0) + q3m.*(eta<0);
Q4 = q4pp.*((xi>0) & (eta>0)) + q4mm.*((xi<0) & (eta<0)) + ...
q4pm.*((xi>0) & (eta<0)) + q4mp.*((xi<0) & (eta>0));
qnew = (1-abs(xi)).*(1-abs(eta)).*q1 + ...
abs(xi).*(1-abs(eta)).*Q2 + ...
abs(eta).*(1-abs(xi)).*Q3 + ...
abs(xi).*abs(eta).*Q4;
qnew = qnew(:);
How can one modify it to solve the two-dimensional advection equation
in a composite domain?
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  1 Comment
Asatur Khurshudyan
Asatur Khurshudyan on 14 Apr 2016
Basically, the way is to use the function qnew twice on Omega_1 and Omega_2 and write proper conditions on the common boundary.

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