Variational inequality transport model on the sphere by the active-set reduced-space algorithm

Abstract

The simulation of the transport problem on the sphere is crucial in the numerical modeling of the transport of trace constituents in atmospheric models. One major issue in the numerical simulation is the ability of the solver to obtain accurate constraint-preserving solutions, i.e., ensuring the predicted solution to stay within the physical range. In this paper, we develop and study a variational inequality (VI) based optimization methodology for constructing a new transport model that naturally satisfies this restriction by removing over- and under-shoots of the solution. With the use of the cubed-sphere mesh, we present an explicit-first-step, single-diagonal coefficient, diagonally implicit Runge–Kutta (ESDIRK) method with an adaptive time step control strategy for the fully implicit temporal integration of the variational inequality problem. And then the resulting nonlinear system arising at each time step is solved by using some nonlinear and linear fast solver technologies, including a variant of inexact Newton methods, i.e., the active-set reduced-space (ASRS) method, and the domain decomposition based preconditioners with a novel analytical Jacobian matrix. A set of tracer transport test cases based on the variational inequality model is presented to demonstrate the efficiency and robustness of the proposed method.