Unconditionally stable, second order, decoupled ensemble schemes for computing evolutionary Boussinesq equations.

In this report we present two unconditionally stable, second order, decoupled ensemble schemes for computing evolutionary Boussinesq equations: the stabilized scalar auxiliary variable Crank-Nicolson leap-frog ensemble scheme (Stab-SAV-CNLF-En) and the stabilized scalar auxiliary variable BDF2 ensemble scheme (Stab-SAV-BDF2-En). The two ensemble schemes adopt the recently developed stabilized scalar auxiliary variable (SAV) idea and ensemble timestepping to achieve both high efficiency and unconditional stability. Specifically, the stabilized SAV approach makes it possible to devise unconditionally long time stable schemes for which the nonlinear terms and coupling terms in the Boussinesq equations are made fully explicit, leading to linear systems with constant coefficient matrices to be solved after spatial discretization. The ensemble timestepping further improves the efficiency by making the coefficient matrices of all realizations the same, so that efficient block solvers can be applied to solve the corresponding one linear system with multiple right hand sides and thus greatly reduce the computational cost. We prove the proposed schemes are unconditionally stable and present implementation details. Ample numerical tests are performed to show the efficiency and effectiveness of the combined approach. [ABSTRACT FROM AUTHOR]