The residual balanced IMEX decomposition for singly-diagonally-implicit schemes

In numerical time-integration with implicit-explicit (IMEX) methods, a within-step adaptable decomposition called residual balanced (RB) decomposition is introduced. This new decomposition maintains time-stepping accuracy even when the implicit equation is only roughly approximated. This novel property is possible because a suitable modification of the traditional IMEX algorithm allows the remaining residual to be seamlessly transferred to the explicit part of the decomposition. The RB decomposition allows an early termination of iterations while preserving time-step accuracy. It can gain computational efficiency by exploring the trade-off between the computational effort placed in the iterative solver and the numerically stable step size. We develop a rigorous theory showing that RB maintains the order of singly diagonally implicit schemes. In computational experiments we show that, in many cases, RB-IMEX reduces the number of iterations when compared with the traditional IMEX method. It is often more stable also. The stability of RB-IMEX is studied using a model containing diffusion and dispersion; in this way, one can visualize how the stability region changes as a function of the number of iterations. Here, computational experiments use ESDIRK schemes for a stiff reaction-advection-diffusion equation, for a Navier-Stokes simulation with acoustic stiffness, and for a semi-implicit implementation of Burguers equation.