Parameter estimation in convection dominated nonlinear convection–diffusion problems by the relaxation method and the adjoint equation

The development of numerical methods for strongly nonlinear convection–diffusion problems with dominant convection is an ongoing topic in numerical analysis. For inverse problems in this setting, there is a need of fast and accurate solvers. Here, we present operator splitting with a Riemann solver for the convective part and a relaxation method for the diffusive part, as a means to achieve this goal. Combined with the adjoint equation method this allows us to solve inverse problems within reasonable time frames and with modest computing power. As an example, the dual-well experiment is considered and the adjoint method is compared with a conjugate gradient algorithm and a Levenberg–Marquardt type of iteration method.