Forward and adjoint sensitivity analysis with continuous explicit Runge–Kutta schemes

We study the numerical solution of tangent linear, first and second order adjoint models with high-order explicit, continuous Runge–Kutta pairs. The approaches currently implemented in popular packages such as SUNDIALS or DASPKADJOINT are based on linear multistep methods. For adaptive time integration of nonlinear models, interpolation of the forward model solution is required during the adjoint model simulation. We propose to use the dense output mechanism built in the continuous Runge–Kutta schemes as a highly accurate and cost-efficient interpolation method in the inverse problem run. We implement our approach in a Fortran library called DENSERKS, which is found to compare well to other similar software on a number of test problems.