simulkade
commented
8 years ago Hi Daniel,
At the beginning, I designed FVTool to solve PDE's with implicit methods. Unfortunately, this design is not convenient when you want to use an explicit integration. Choose a time step using the CFL condition: https://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition
I keep this issue open to find a better way of implementing Explicit methods. Most probably, I will write a function called phi_new=solveExplicitPDE(m, M_transient, RHS, BC). I really have to spend some time on it.
Whatsoever
Dear Dr. Eftekhari, I have another question which is probably kind of silly, although it could be quite helpful to me. If I want to solve a classical transport advection equation (without sources), I could end up with the following discrete system:
where E is a diagonal matrix with the values of the porosity C_n1 is the vector of dependent variable at time n+1 C_n is the vector of dependent variable at time n Dt is the time step L is a (probably tridiagonal matrix) matrix related to the spatial discretization for advection and diffusion. thetha is a factor from 1 to 0, when 1 forward euler and when 0 implicit euler. q is a vector of boundary conditions
In the propose discretization the spatial discretization order is not clear but the temporal is obviously first order (although if thetha is 1/2 it will be second order). I would like to work with a time explicit scheme, namely forward euler and thetha 1. First, for a 1D problem
In the tutorial folder, most of the problems in 1D are solved by an implicit approach, except (I think) ' diffusiontutorialExplicit .m'. Where the solution is given by the line:
% step 1: calculate divergence term RHS = divergenceTerm(Dave._gradientTerm(c_old)); % step 2: calculate the new value for internal cells c_old_int=internalCells(c_old); c_val_int = dt_excludeGhostRHS(meshstruct, RHS+constantSourceTerm(c_old))+c_old_int(:);
I have tried to add advection with the following modification:
% step 1: calculate divergence term RHSd = divergenceTerm(Dave._gradientTerm(c_old)); RHSv = divergenceTerm(u._arithmeticMean(c_old)); % step 2: calculate the new value for internal cells c_old_int=internalCells(c_old); c_val_int = subdt*excludeGhostRHS(m, -RHSv+RHSd+constantSourceTerm(c_old))+c_old_int(:);
Although the results are not satisfactory, I wonder if it is because the time step is big, the implementation that I use is wrong or maybe something to do with boundary conditions? Which ways there are to use a simple forward euler once the spatial discretization has been done with FVT tool?
Thanks again
Best, Daniel