Have a look to this paper, it illustrates the non-linear source dominated transport problem

http://scholar.google.it/scholar_url?url=https://digitalcommons.unl.edu/cgi/viewcontent.cgi%3Farticle%3D1282%26context%3Dnasapub&hl=it&sa=X&scisig=AAGBfm1k1ZpWAWj8oB9nPAchdcPpUiZNDQ&nossl=1&oi=scholarr

Hi Omar,

have you checked the book "Solving Direct and Inverse Heat Conduction Problems" by Jan Taler and Piotr Duda, it might be helpful.

Best of luck!

Best regards,

Antoni Artinov

Many years ago we numerically investigated pressure waves using the MacCormack algorithm which worked pretty well for one dimensional problems where the first coordinate is orientated perpendicular to the pressure front. (Have a look to "some remarks" of the article "MacCormack method" in Wikipedia regarding the reversed direction of integration.) But we were massed in terrible numerical problems when adding the second dimension.

Good luck!

Nikolaus Hannoschöck

Have you tried using standard methods for the convection and diffusion terms, then lagging the source term?

Dear Prof. Denaro, Thanks for your hint to an interesting paper on hyperbolic systems with stiff source terms. Unfortunately Google tries to keep us trapped in Google by encrypting the links. The original link is quite short: https://digitalcommons.unl.edu/nasapub/282/

One possibility is use operator-splitting (second order Strang splitting) to split the linear- and nonlinear parts Then the linear (convention-diffusion part) can be solved by an implicit method leading to a non-symmetric linear system. The non-part are then a collection of (indepedent) stiff ODE-equations for which there are a wide variety of methods. The critical step is wether the opererator splitting will wil converge properly. This paper may be useful: Article An Analysis of Operator Splitting Techniques in the Stiff Case

ADI can also be used in 2D, e.g. see https://orbit.dtu.dk/files/53411889/AbstractMultiCoreChallenge2012.pdf

I recommend the projection method as follow:

1-split the equation to two fractional steps (i.e diffusion and convection equations)

2- rewrite convection equation as it's conservative form

3-discrete spatial domain on a staggered grid

4-set an appropriate initial and boundary condition for a parameter (i.e Phi) you would like to transport in spatial domain by current fractional step

5-rewrite convection equation in discretized form as follow:

(phi_i^(n+1)) x (deltaX) = (phi^(n)) x (deltaX)-(phi_OUT^(n/n+1)) +(phi_IN^(n/n+1))

i: cell index

n: time step

_OUT = index of out flow @ I+1, index of out flow boundary of cell

_IN = index of inflow @ I-1, index of inflow boundary of cell

phi_OUT^(n/n+1) = u_(i+1)^(n) *Delt*(phi_(i)^(n))

phi_IN^(n/n+1) = u_(i-1)^(n) *Delt*(phi_(i-1)^(n))

u: fluid velocity

in staggered grid first advect and diffuse scalar parameter in one direction after that consider perpendicular direction

aforementioned discretize of convection equation is similar to first order backward FDM, one can use high order spatial discretization

6- solve diffusion question in relative direction

7- discretize second order spatial derivation by central FDM

8- discretize time dependent value as phi = teta x (phi^(n+1)) +(1-teta)* phi^(n)

9- set teta=0.5 crank nicolson

10-use ADI method for matrix solution

11-do 2:10 for perpendicular direction