# Calculating the species mass consumption from implicit reaction-term in diffusion-reaction equation

The 1D diffusion equation with a chemical source term has the following form: $$\frac{\partial Y}{\partial t} = D \frac{\partial^2 Y}{\partial x^2} - k Y,$$ where $$Y$$ is the molar concentration of the diffusing species (mol/m³), $$D$$ is the diffusion coefficient (m²/s), and $$k$$ is the reaction rate coefficient (1/s).

The boundary conditions are: $$\frac{\partial Y}{\partial x}|_{x=0} = 0,$$ $$-D \frac{\partial Y}{\partial x}|_{x=xN} = mtc (Y_c - Y),$$ where $$Y_c$$ is the species concentration outside the simulation domain (mol/m³), and $$mtc$$ is the mass transfer coefficient (m/s).

I can solve this equation numerically stable using an implicit approach: $$Y(t=t_0) = A\cdot Y(t=t_0+dt),$$ where the matrix $$A$$ has a term that depends on $$k$$ in the diagonal only.

My Problem is the following: I need to know how much species mass was converted by the chemical reactions at each grid-point, so I can calculate the heat generated/consumed at each grid-point as well as know how much of an additional stationary solid phase was converted in the process. As I solve the equation implicitly I do not know how to split the diffusive contribution and the reactive contribution to the final $$Y$$ at each grid-point. It is further complicated by mass not being conserved in the system due to the flux at the right BC.

Without reactions I can always calculate the difference of $$Y$$ before and after a time step and I know how much mass was transported in total to or from this grid-point in the given time. If I add the reaction term then this difference is a combination of mass being converted due to the reaction and mass being transported by diffusion.

I have spent some time researching reaction-diffusion equations and their solutions, however, I have not been able to find any information about how to calculate how much of my species was converted chemically at each grid-point.

The only workaround I have found so far was to explicitly calculate a chemical source term using the initial value of $$Y(t=t_0)$$ at a given time-step and then including it as an explicit source in the above equation. The problem with this approach was that this leads to me having to severely limit the size of the time-step such that only a fraction of the available mass at a grid-point is converted in each time step. However, this leads to unsustainably small time-steps for some examples I encountered.

I would be grateful for any help regarding this issue. I doubt there is an exact way to split the contributions of diffusion and reaction and am happy for workarounds or approximations of all sort.