Objective: I am trying to numerically solve $C(x,y,t)$ from the following advection-diffusion-reaction equation in 2D space (x,y) and time. I will be testing my numerical solution with an approximate 1D analytical solution (given on 2nd page in this file). However, the issue is that I am not able to get correct numerical solution.$$\begin{align}\text{ADR Equation: }\frac{\partial C}{\partial t} + \nabla\left(v  C - D\nabla{C} \right)=  \alpha C\end{align}$$Method: I discretized the above ADR equation in 2D using finite-difference implicit scheme and I get the following equation in difference form.$$\begin{align}p_1C^{n+1}_{i,j-1}+p_2C^{n+1}_{i-1,j}+p_3C^{n+1}_{i,j}+p_4C^{n+1}_{i+1,j}+p_5C^{n+1}_{i,j+1} = C^{n}_{i,j}\end{align}$$I want to solve this as a system of equations using $A^{n+1} C^{n+1}=C^{n}$ Variables: $A$ is a matrix formed from variables $p_1, p_2, p_3, p_4, p_5$, which are constants in time but vary along $(x,y)$ grid. Superscript $n$ represents time index and subscripts $i,j$ represent space along $x,y$. I.C: $$\textrm{$C(x,y,0) = C_i$, assume $C_i$ as 1}$$B.C: $$\frac{\partial C}{\partial x} = \frac{\partial C}{\partial y} = 0 \text{ at boundaries}$$. $$C(0,y=1,t) = \begin{cases} C_0\quad & \text{if $0<t\leq t_0$} \\  0 \quad & \text{if $t>t_0$} \end{cases}$$ $\textrm{assume $C_0$ and $t_0$ as 5000 and 2 days, respectively}$Numerical Solution: I estimated $C(x,y,t)$ for $t=0$ to $t=100$ days, where dimensions of $x$ and $y$ are shown below in results. $$\textrm{Fig-1: Snapshots of $C(x,y)$ at various $t$}$$ In Fig-2, I am showing $C(x,t)$ for $0<t\leq t_0$ and $t=0$ to $t=100$ days:$$\textrm{Fig-2: Snapshots of $C(x,t)$ obtained after averaging $C(x,y,t)$ over Y-axis}$$Analytical Solution: Here I am showing $C(x,t)$ for $0<t\leq t_0$ and $t=0$ to $t=100$ days. $$\textrm{Fig-3: Snapshots of $C(x,t)$. I am trying to match this with Fig-2}$$Issue: I have been trying to figure out if there is a problem in my numerical solution developed as per the implicit scheme discussed above; however, I have spent several hours and I couldn't figure it out. Please let me know if I missed providing anything.
Not getting correct numerical solution for Advection-Diffusion-Reaction eqn
finite difference;advection diffusion;implicit methods;diffusion