An adaptive local grid refinement based on the exact peak capture and oscillation free scheme to solve transport equations

An adaptive local grid refinement (ALGR) algorithm based on exact peak capture and oscillation free scheme (EPCOF) is developed to solve transport equations. This algorithm consists of the Lagrangian-Eulerian decoupling of advection-diffusion transport, backward-node tracking, forwardnode tracking, and adaptive local grid refinement based on exact peak capture and oscillation free strategies. Means of checking accumulated mass-balance errors are provided. Application of the algorithm to four benchmark (two 1-D and two 2-D) problems under a variety of conditions indicated that it completely eliminated peak clipping, spurious oscillation, numerical diffusion, and grid-orientation difficulties. It yielded identical results, within the error tolerance, to exact solutions for 35 of the 43 test cases; very good solutions, albeit not exact to within the error tolerance, were obtained for the remaining 8 cases. Accumulated mass-balance errors are very small for all cases with the maximum error of less than 1%. The proposed ALGR-EPCOF is also used to simulate a three-dimensional advective-diffusive-reactive transport problem. Simulation results are accurate to within error tolerance in comparison to exact solutions for the case of advective-reactive transport. This demonstrates that the use of tetrangulating the activated forward-tracked nodes is a promising one. The ALGR-EPCOF approach could solve the advective-reactive transport problems exactly, within any prescribed error tolerance, using mesh Peclet numbers ranging from 0 to infinity and very large mesh Courant numbers in the Lagrangian-step computation. The size of the mesh Courant number is limited only by the accuracy requirement of the diffusion solver in the Eulerian-step computation. If the associated diffusion solver can solve the diffusion transport exactly within the same error tolerance, then the ALGR-EPCOF approach can solve the advection-diffusion-reactive problems exactly to within the prescribed error tolerance.