Recovering the spatial distribution of a groundwater contaminant at a prior time: The forward collocation method

We study the backward parabolic problem related to the convection-diffusion operator Au:≡ut−(D(x)ux)x+(c(x)u)x when the diffusion coefficient D(x)D(x) may be discontinuous. The forward collocation method (FC-method) is used for numerical solution of this backward transmission problem. According to the method, we approximate the unknown function ϕ(x)=u(x,t0)ϕ(x)=u(x,t0) by the piecewise linear continuous, Lagrange type of basis functions. Moreover, we solve the obtained ill-conditioned system of algebraic equations by using truncated singular value decomposition (TSVD).