Long time unconditional stability of a two-level hybrid method for nonstationary incompressible Navier–Stokes equations

A three‐level time‐split MacCormack method for two‐dimensional nonlinear reaction‐diffusion equations

“…This convergence rate is confirmed by a large set of numerical experiments (see both Figures 1‐3 and Tables 1‐6). Numerical evidences also show that the new algorithm is (a) more efficient and effective than a wide range of numerical approaches 4,5,9,16,19,25-27,30 and (b) fast and robust tools for the integration of general systems of parabolic PDEs. However, the time‐split MacCormack method is not a suitable approach for solving high Reynolds number flows where the viscous region becomes very thin.…”

An efficient three‐level explicit time‐split scheme for solving two‐dimensional unsteady nonlinear coupled Burgers' equations

“…where we have also absorbed the first-order term into the error term O(k + k 2 h −1 ). Using approximation (30), it is too simple to observe that…”

“…Specifically, the hybrid version of MacCormack has been used to solve the mixed Stokes‐Darcy and two‐dimensional time‐dependent incompressible Navier‐Stokes equations, whereas the three‐level time‐split MacCormack was applied to two‐dimensional time‐dependent reaction‐diffusion, heat conduction, convection‐diffusion equations, and linear/nonlinear convection‐diffusion‐reaction equations with constant coefficients (diffusive term equals 1 and convective velocity in the range: −1, 0.8, and 1). The analysis has suggested that the three‐level explicit time‐split MacCormack is fast, second‐order convergent in time, and fourth‐order accurate in space . We recall that the three‐level time‐split applies to a time‐dependent problem of the form u t = A 1 ( u )+ A 2 ( u ), where A j ( j =1,2) are differential operators so that each subproblem u t = A j ( u ), j =1,2, is solved independently using the original MacCormack approach.…”

“…The analysis has suggested that the three-level explicit time-split MacCormack is fast, second-order convergent in time, and fourth-order accurate in space. 9,[28][29][30][31][32][33][34][35][36] We recall that the three-level time-split applies to a time-dependent problem of the form u t = A 1 (u) + A 2 (u), where A j ( j = 1, 2) are differential operators so that each subproblem u t = A j (u), j = 1, 2, is solved independently using the original MacCormack approach.…”

A Two-Level Fourth-Order Approach For Time-Fractional Convection-Diffusion-Reaction Equation With Variable Coefficients

“…In the literature the big challenge with such equations is the design of efficient and accurate numerical methods and computational cost is the main issue to be considered for any numerical scheme. For classical integer order ordinary or partial differential equations (ODEs/PDEs) such as: systems of ODEs, Navier-Stokes equations, mixed Stokes-Darcy model, shallow water equations, advection-diffusions problems, convection-diffusionreaction equations, heat conduction [22,24,52,40,6,28,30,31,34,14,42,35,37,10,27], a wide class of numerical approaches have been deeply analyzed: finite difference techniques, two-level MacCormack procedure, spectral methods, full implicit finite difference schemes, two-level factored approaches, compact ADI methods and multi-level finite difference formulations. For more details, we refer the readers to [7,11,33,32,36,49,15,38,13,38,23,44,26,25,45,39,46,29] and references therein.…”