A time-centered split for implicit discretization of unsteady advection problems

Environmental flows (e.g. river and atmospheric flows) governed by the shallow water equations (SWE) are usually dominated by the advective mechanism over multiple time-scales. The combination of time dependency and nonlinear advection creates difficulties in the numerical solution of the SWE. A fully-implicit scheme is desirable because a relatively large time step may be used in a simulation. However, nonlinearity in a fully implicit method results in a system of nonlinear equations to be solved at each time step. To address this difficulty, a new method for implicit solution of unsteady nonlinear advection equations is developed in this research. This Time-Centered Split (TCS) method uses a nested application of the midpoint rule to computationally decouple advection terms in a temporally second-order accurate time-marching discretization. The method requires solution of only two sets of linear equations without an outer iteration, and is theoretically applicable to quadratically-nonlinear coupled equations for any number of variables. To explore its characteristics, the TCS algorithm is first applied to onedimensional problems and compared to the conventional nonlinear solution methods. The temporal accuracy and practical stability of the method is confirmed using these 1D examples. It is shown that TCS can computationally linearize unsteady nonlinear advection problems without either 1) outer iteration or 2) calculation of the Jacobian. A family of the TCS method is created in one general form by introducing weighting factors to different terms. We prove both analytically and by examples that the value of the weighting factors does not affect the order of accuracy of the scheme. In addition, the TCS method can not only computationally linearize but also decouple an equation system of coupled variables using special combinations of weighting factors. Hence, the TCS method provides flexibilities and efficiency in applications.