How are RK methods for solving time-dependent PDEs implemented?
I am trying to reproduce results of a thesis. It is a advection-diffusion unsteady equation. It is clearly mentioned that they have used RK method for time integration. I cannot use finite difference methods like Crank-Nicolson or any other method like that.
Thanks.
The conventional way of treating time-dependent PDE is to employ a method-of-lines like procedure [A], [B]. Although that approach was originally only referred to employing finite-difference for the spatial terms, the name persisted and is now used for methods that employ a particular method (Finite Difference/Element/Volume, Discontinuous Galerkin, ...) for the spatial derivatives and something different for the temporal terms.
In particular, for the advection diffusion equation
$$ \partial_t c = \nabla \cdot (D \nabla c) + a \cdot \nabla c + s$$ you would apply e.g. Finite Element for the spatial derivatives, i.e., $$\nabla \cdot (D \nabla c), \quad a \cdot \nabla c$$
which gives you then a semi-discretized system like
$$ \frac{d \boldsymbol C}{dt} = \boldsymbol F\big(\boldsymbol C(t) \big) $$
for which you can then use (in principle) any ODE solver, given an initial condition $\boldsymbol C(t_0) = \boldsymbol C_0$.
[A] Application of the Method of Lines to Parabolic Partial Differential Equations With Error Estimates
[B] W. E. Schiesser. The numerical method of lines: integration of partial differential equations