In the article, we propose a combination method based on the multigrid method and constraint data to solve the inverse problem in the context of the nonlinear convection–diffusion equation in the multiphase porous media flow. The inverse problem consists of a data-fitting term involving the discretization of a direct problem, a constraint term concerning the incorporation of constraint data, and a regularization term dealing with the improvement of stability. A multigrid method, which is specialized for large-scale problems and works by keeping the consistence of objective functionals between different grids, is applied in the process of inversion. Based on the numerical results, the proposed combination method has the advantages of fast calculation, high precision, good stability, and strong anti-noise ability in computation. It obtains good performance under various noise levels, as well as outperforming any one method used alone.

This study presents an efficient method using the local radial basis function finite difference scheme (RBF-FD). The innovative coefficients are derived from the integrals of the multiquadric (MQ) function. Theoretical convergence rates for the coefficients used in function derivative approximation are provided. The proposed scheme utilizes RBF-FD estimations on three-point non-uniform stencils to construct the final approximation on a tensor grid for the 3D Heston–Hull–White (HHW) PDE, which is relevant in economics and mathematical finance. Numerical evidence and comparative analyses validate the results and the proposed scheme.