Manufactured solution to 2d convection-diffusion with homogeneous Robin boundary conditions

I am looking for a manufactured (or analytical if it exists) solution to the 2d boundary-value problem

$$\frac{\partial u}{\partial t} = \mathbf{a} \cdot \nabla u + D \nabla^2 u \quad \quad \mbox{in } \Omega$$ $$u \ \mathbf{a} \cdot \mathbf{\hat{n}} + D \ \nabla u \cdot \mathbf{\hat{n}} = 0 \quad \quad \mbox{on } \partial \Omega$$

where $$\Omega$$ is a rectangular domain and $$\mathbf{a}$$ and $$D$$ are independent of space and time.

I would highly appreciate any help / suggestion / reference on the matter.

• You mean like $u\equiv 0$? Otherwise replace $L[u]=0$ in $Ω$, $R[u]=0$ on $∂Ω$ with $L[u]=L[p]$, $R[u]=R[p]$ for any sufficiently smooth function $p$, obviously $u=p$ is the reference or manufactured solution. Oct 1 '21 at 13:22
• But you need a source term to balance your manufactured solution. Oct 2 '21 at 14:06

Usually manufactured solutions are used to verify a solver. As stated in the comment section, you should consider a source term both in the domain $$\Omega$$ and on the boundary $$\partial \Omega$$
$$\frac{\partial u}{\partial t} - {\bf a} \cdot \nabla u - D \nabla^2 u = f({\bf x},t) \quad \text{in \Omega} ,$$ $$u {\bf a} \cdot {\bf \hat{n}} + D\nabla u \cdot {\bf \hat{n}} = g({\bf x}, t) \quad \text{on \partial \Omega}.$$
Here, you simply plug any $$u$$ of your choice, then you compute the source terms $$f$$ and $$g$$ by hand, and then you can test the consistency of your solver by comparing the analytical $$u$$ you choose and the numerical $$u_h$$ you have computed. Once you do it, you are sure that you solver works also on the original equation, i.e., by setting $$f=g=0$$.
For instance, if you plug $$u = e^{-t}\sin(x)\cos(y)$$ you get (double check!)
$$f({\bf x},t) = e^{-t} (\cos(y) \sin(x) - 2 D \cos(y) \sin(x) + a_x \cos(x) \cos(y) - a_y \sin(x) \sin(y)),$$
$$g({\bf x},t) = e^{-t} (D n_x \cos(x) \cos(y) - D n_y \sin(x) \sin(y) + a_x n_x \cos(y) \sin(x) + a_y n_y \cos(y) \sin(x)).$$