A New Algorithm with Square Conservative Exponential Integral for Transport Equation

Abstract

AbstractThe square conservative exponential integral method (SCEIM) is proposed for transport problems on the sphere. The method is a combination of the square conservation algorithm and the exponential integral method. The main emphasis in the development of SCEIM is on conservation, positive-definite, and reversibility as well as achieving comparable accuracy to other published schemes. The most significant advantage of SCEIM is to change the forward model to the backward model by setting a negative time step, and the backward model can be used to solve the inverse problem. Moreover, the polar problem is significantly improved by using a simple effective central skip-point difference scheme without major penalty on the overall effectiveness of SCEIM. To demonstrate the effectiveness and generality of the SCEIM, this method is evaluated by standard cosine bell tests and deformational flow tests. The numerical results show that SCEIM is a time-convergence method as well as a grid-convergence method, and has a strong shape-preserving ability. In the tests of the inverse problem, the sharp fronts are successfully regressed back into their initial weak fronts and the cosine bells move against the wind direction and return to the initial position with high accuracy. The numerical results of forward simulations are compared with those of published schemes, the total mass conservation, and error norms are competitive in term of accuracy.