I have 2 nonlinear parabolic PDEs which are coupled (heat and mass diffusion equation). I am looking for a method with good convergence and stability. The language I should write my code in is FORTRAN.
The simplest approach would be to discretize the equations using finite difference in space and then use an implicit time stepper for the time discretization.To solve the arising non-linear systems I would use a 'Lapack' non-linear solver in Fortran.
Dear Behnam Dastvareh,
See link: https://www.researchgate.net/publication/261729269_Numerical_Method_in_Meteorology_Solved_Problems
Regards,
Milivoj B. Gavrilov
Book Numerical Method in Meteorology: Solved Problems
This is a good question with many possible answers. In addition to the helpful answers already given, there is a bit more to add.
Another good place to look for the numerical solution of a coupled system of non-linear PDEs is Section 2.3 in
http://www4.ncsu.edu/eos/users/s/shearer/www/papers/PHYS_D_01.pdf
See also Section 5 in the same paper.
More to the point, see Section 6.2 in
A Method for the Spatial Discretization of Parabolic Equations in One Space Variable
https://www.researchgate.net/publication/244958749_A_Method_for_the_Spatial_Discretization_of_Parabolic_Equations_in_One_Space_Variable
Article A Method for the Spatial Discretization of Parabolic Equatio...
You can use Perturbation Methods for Engineers and Scientists (CRC Press Library of Engineering Mathem) 1st Edition
by Alan W. Bush on pages 182 and 183, you can find the FORTRAN program to solve the Heat equation mass continuity, X momentum, Y momentum
Thanks for all your response.
Pietro: The LAPACK contains the routines only for linear algebra. Would you please write the name of the routine here?
For example look at quasi.f in http://www.netlib.org/napack/. But I bet you could find also something else.
Discretize the space domain and express all spatial derivatives using finite difference method, however do not discretize the time domain. This will give you a set of nonlinear ordinary differential equations corresponding to solution at the grid points. Use any ODE solver to solve the set of nonlinear ODE's.
Thaks Saroj. Yes I am using the method of line to solve the problem. I found it the most reliable method for this type of problems. One can use DLSODA solver from netlib to solve the resulting system of ordinary differential equations in time.
Dear Behnam,
There is nothing like "the best, most stable ... method", it is a matter of choosing between advantages and disadvantages. I would do first a discretization in time (e,g, Euler implicit) and end up at any time step with coupled systems of nonlinear elliptic equations. Then I would linearie them, or better said define a linear iterative method, generating a system of linear elliptic equations at each iteration step. Next you discretise and solve numerically these equations by using the method that suits you best.
An example in this sense is our recent publication
https://www.researchgate.net/publication/288713617_A_convergent_mass_conservative_numerical_scheme_based_on_mixed_finite_elements_for_two-phase_flow_in_porous_media
(see also https://www.researchgate.net/publication/273579720_A_robust_linearization_scheme_for_finite_volume_based_discretizations_for_simulation_of_two-phase_flow_in_porous_media)
The method is a contraction type scheme and converges for any initial guess, but it is only linear. The method can be easier understood from
https://www.researchgate.net/publication/280590357_A_study_on_iterative_methods_for_solving_Richards_equation
or from section 4.1 in
https://www.researchgate.net/publication/239843943_Convergence_analysis_for_a_conformal_discretization_of_a_model_for_precipitation_and_dissolution_in_porous_media
If you want to read about the Newton method in the context of nonlinear evolution problems I suggest
https://www.researchgate.net/publication/283998274_Newton_Type_Methods_for_the_Mixed_Finite_Element_Discretization_of_Some_Degenerate_Parabolic_Equations
Good luck,
Sorin
Article A convergent mass conservative numerical scheme based on mix...
Article A robust linearization scheme for finite volume based discre...
Article A study on iterative methods for solving Richards` equation
Article Convergence analysis for a conformal discretization of a mod...
Chapter Newton—Type Methods for the Mixed Finite Element Discretizat...
you may send me ur problem. I may have a look with my code in matlab, if you want.. later it may help you
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