nonlinear tutorial

Author: simulkade

Examples/Tutorial/diffusionNonlinearTutorial.m

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+% Nonlinear diffusion equation: a tutorial
+% see http://fvt.simulkade.com/posts/2015-04-06-solving-nonlinear-pdes-with-fvm.html
+L= 1.0; % domain length
+Nx= 100; % number of cells
+m= createMesh1D(Nx, L); % create a 1D mesh
+D0= 1.0; % diffusion coefficient constant
+% Define the diffusion coefficientand its derivative
+D=@(phi)(D0*(1.0+phi.^2));
+dD=@(phi)(2.0*D0*phi);
+% create boundary condition
+BC = createBC(m);
+BC.left.a(:)=0.0;
+BC.left.b(:)=1.0;
+BC.left.c(:)=5.0;
+BC.right.a(:)=0.0;
+BC.right.b(:)=1.0;
+BC.right.c(:)=0.0;
+[Mbc, RHSbc]= boundaryCondition(m,BC);
+% define the initial condition
+phi0= 0.0; % initial value
+phi.Old= createCellVariable(m, phi0, BC); % create initial cells
+alfa=createCellVariable(m, 1.0);
+phi.value= phi.Old;
+% define the time steps
+dt= 0.001*L*L/D0; % a proper time step for diffusion process
+for i=1:10
+    err=100;
+    [Mt, RHSt] = transientTerm(m,alfa,dt, phi);
+    while err>1e-10
+        % calculate the diffusion coefficient
+        Dface= harmonicMean(m,D(phi.value));
+        % calculate the face value of phi_0
+        phi_face= arithmeticMean(m,phi.value);
+        % calculate the velocity for convection term
+        u= funceval(dD, phi_face).*gradientTerm(m,phi.value);
+        % diffusion term
+        Mdif= diffusionTerm(m,Dface);
+        % convection term
+        Mconv= convectionTerm(m,u);
+        % divergence term on the RHS
+        RHSlin= divergenceTerm(m,u.*phi_face);
+        % matrix of coefficients
+        M= Mt-Mdif-Mconv+Mbc;
+        % RHS vector
+        RHS= RHSbc+RHSt-RHSlin;
+        % call the linear solver
+        phi_new= solvePDE(m, M, RHS);
+        % calculate the error
+        err= max(abs(phi_new(:)-phi.value(:)));
+        % assign the new phi to the old phi
+        phi.value=phi_new;
+    end
+    phi.Old= phi.value;
+    visualizeCells(m,phi.value); drawnow;
+end