Random walk on spheres method for solving anisotropic drift-diffusion problems

A The reciprocity theorem

In this appendix, we prove the reciprocity relation which we use in the algorithm for calculation of fluxes. Thus we deal with the direct (4.1) and adjoint (4.2) equations in the domain G with Dirichlet boundary conditions w d ⁢ ( 𝐲 ) = 0 , 𝐲 ∈ Γ = ∂ ⁡ G and w a ⁢ ( 𝐲 ) = 1 if 𝐲 ∈ Γ 1 and w a ⁢ ( 𝐲 ) = 0 if 𝐲 ∈ Γ 2 . For simplicity we consider here the case Γ = Γ 1 ∪ Γ 2 , which can be easily generalized to the general case.

Let us multiply the direct equation (4.1) by w a :

and the adjoint equation (4.2) by w d :

Then we subtract one from another and take the integral over the volume G . As a result we get

The second volume integral is reduced to

By applying the Gauss–Ostrogradsky formula, the volume integrals are transformed to surface integrals

where cos ⁡ γ i are the direct cosines of vector 𝐱 . Due to the boundary conditions for w d and w a , only the term containing the function w a ⁢ ( 𝐱 ) = 1 where 𝐱 ∈ Γ 1 is nonzero. After all reductions we get

that is, a flux of particles to the boundary Γ 1 . Here cos ⁡ γ i is the outside direct cosine of the normal vector.