Background

Today, we will solve a system of convection-diffusion-reaction equations,

$$\begin{array}{rcl} \dot{u}_1 + b \cdot \nabla u_1 - \nabla... ... u_2(\cdot,0) &=& u_{20} \quad \mbox{in } \Omega, \end{array}$$ (1)

where we note that the right-hand side depends on the solution $u = (u_1,u_2)$ itself. This is a system of nonlinear equations (if $f_1$ or $f_2$ are nonlinear). Typically, $f_1$ and $f_2$ are chosen to model a chemical reaction.

The $\mathrm{dG}(0)$ formulation is given by

$$\begin{array}{ll} \int_{\Omega} U_{1,n} v \, dx + k \int_... ...Omega} U_{2,n-1} v \, dx \quad \forall v \in V_h, \end{array}$$ (2)

where $k = t_n - t_{n-1}$ denotes the size of the time step, $U_{1,n}$, $U_{2,n}$ denote the values at $t=t_n$, and $U_{1,n-1}$, $U_{2,n-1}$ denote the values at $t=t_{n-1}$.

Before today's computer session, make sure that you understand and can answer the following questions.

Question 1 Derive the variational formulation (2) from the system of equations (1).

Question 2 How does the corresponding variational formulation look for the $\mathrm{cG}(1)$ method? This method is also known as the Crank-Nicolson method.

Question 3 Assume that we have an (irreversible) chemical reaction of the form $A + B \rightarrow C$, let $u_1(x,t)$ be the concentration of the substance $A$ and let $u_2(x,t)$ be the concentration of the substance $B$. Can you motivate that this is modeled by the system (1) if we take

$$\begin{array}{rcl} f_1(u_1,u_2,x,t) &=& - c u_1 u_2, \\ f_2(u_1,u_2,x,t) &=& - c u_1 u_2. \end{array}$$ (3)

The constant $c$ determines the rate of the reaction and is called the reaction constant.