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 A u : ≡ u t − ( D ( x ) u x ) x + ( c ( x ) u ) x when the diffusion coefficient 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 , t 0 ) 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).