Reaction-diffusion equation terms

Question

I have been given the reaction diffusion equation:
$\frac{\partial u}{\partial t} = \kappa\frac{\partial^2 u}{\partial x^2} + u - u^3$
with neumann boundaries and in initial equation for the starting time. I have only worked with the diffusion equation before, but the question asks for the theta method for the diffusion term and the euler method for the reaction term. I don't understand which part is the reaction term and which is the diffusion term and how to separate my discretisation. Any help or pointers would be greatly appreciated :)

Answer 1

In $$ u_t=κu_{xx}+cu_x+f(u) $$ the first term on the right is the diffusion term, the second one the transport term and the last one the reaction term.

The method mentioned is called "operator splitting". Essentially, you solve alternatingly $u_t=κu_{xx}$ and $u_t=f(u)$ with the given methods and the same step size. One can employ variations like dividing the diffusion propagation into half-steps before and after the reaction propagation. One would need to make a more quantitative error analysis to see if that brings any advantages, and for what time and space step sizes the order 1 method in the slow diffusion produces about the same amount of error as the order 2 method for the faster reaction.

Answer 2

According to wikipedia, your equation is called after Newell, Whitehead, and Segel. Searching for that equation on googlescholar, you will find many papers that solve it analytically and numerically, see for instance paper1, paper2, paper3.

Concerning your question about the naming of the terms, I think it is clearer if you write the reaction-convection-diffusion equation as \begin{equation} \partial_t c=\nabla\cdot(D\nabla c)-\nabla\cdot(\mathbf{v} c)+R \end{equation} where $c$ stands for some scalar quantity like concentration or temperature, where $D$ is the associated diffusivity, where $\mathbf{v}$ is the velocity of some background fluid, and where $R(c)$ is the reaction term. When the diffusion constant is spatially invariant, and for incompressible fluids $(\nabla\cdot\mathbf{v}=0)$, and for 1d systems, the above equation simplifies to \begin{equation} \partial_t c=D\partial_x^2 c-v\partial_x c+R \end{equation} which is not of the form of your equation.