Partitioned Time Stepping Method For A Parabolic Two Domain Problem

It is necessary to deal with the coupling system of multi region with multi model in many practical applications.That is to say,different sub-regions have different PDE models or different sub-regions have the same model but different parameters.A complete coupling system is formed by applying appropriate interface conditions on the interface between the two subregions.In this paper,we study the interaction between the atmosphere and the ocean.Mathematically,it is described by a coupled system of two parabolic equations with appropriate interface conditions.The coupling model is more complex than the single model.If the whole system be solved directly,we will face the difficulties of large scale computation and inconvenient programming.In order to solve this problem and make full use of the existing algorithm of a single sub-model,we hope to find a decoupling algorithm to separate the coupling problem into two independent sub-problems,and then solve them separately.For a two heat equations coupling system,firstly,the backward Euler scheme is used to discretize the time,and the terms on the interface are treated explicitly.The effectiveness and convergence of the two existing first-order decoupling algorithms are verified numerically.Secondly,two second-order decoupling methods are proposed: BDF2 scheme is the combination of a second-order backward differentiation formula and the second-order Gear's extrapolation approach,AMB2 scheme is a combination of the second-order Adams Moulton and second-order Adams-Bashforth methods.Both schemes only require the solution of two decoupled parablic problems at each time step.These decoupling methods not only reduce the calculation scale,but also realize the flexible selection of appropriate algorithm to solve different subproblems.Hence,they are very efficient and can be easily implemented by using legacy codes,which reduce the difficulty of reconstructing algorithm and programming.Theoretically,we establish the unconditional stability and the error estimation for both schemes.Besides,the numerical tests also confirm the corresponding theoretical results.