Continuous PDE to discrete solution 1. model of pysical process (PDE) 2. discretize spatial domain into nodes/cells/elements 3. choose schemes for advection, diffusion, source, pressure, etc terms 4. derive semi-discrete ODE describing evolution of state vector 5. choose schemes for evolving the discrete equation (implicit, explicit, IMEX, ADI, etc) 6. choose solvers for solving the resulting linear / nonlinear system of equations Example for 2D tracer advection-diffusion equation $\frac{ \partial \rho }{ \partial t } + (u \cdot \nabla)\rho = \nabla(\kappa \nabla)\rho + F_{\rho}(x,t)$ $u = \begin{bmatrix}u \\ d\end{bmatrix}$, $x = \begin{bmatrix}x \\ y\end{bmatrix}$ $\rho$ → ${} \hat{\rho}(t) {}$ Advection: $A(\rho)$ → $\underline{A}(\hat{\rho})$ (QUICK) sometimes becomes nonlinear Diffusion: $D(\rho)$ → $\underline{D}(\hat{\rho})$ (CDS) this should generally be implicit because it's the most unstable $\frac{d\hat{\rho}}{dt}+\underline{A}(\hat{\rho})=\underline{D}(\hat{\rho})+\underline{F}(\hat{\rho})$ $\frac{\hat{\rho}^{n+1}-\hat{\rho}^{n}}{\Delta t}+\underline{A}^{n}(\hat{\rho}^{n})=D^{n+1}(\hat{\rho}^{n+1})+F^{n}(\rho^{n})$ $\underbrace{ (I-D^{n+1}) }_{ A }\underbrace{ \hat{\rho}^{n+1} }_{ x }=\underbrace{ \hat{\rho}^{n}+\Delta t(\underline{A}^{n+1}(\hat{\rho}^{n+1}))+\Delta tF^{n}(\rho^{n}) }_{ b }$