We introduce a technique that makes every explicit Runge--Kutta (ERK) time stepping method invariant domain preserving and mass conservative when applied to high-order discretizations of the Cauchy problem associated with systems nonlinear conservation equations. The key idea is at each stage ERK scheme one computes low-order update, both defined from same intermediate stage, then applies nonli...

Singular source terms in sub-diffusion equations may lead to the unboundedness of solutions, which will bring a severe reduction convergence order existing time-stepping schemes. In this work, we propose two efficient schemes for solving with class mildly singular time. One discretization is based on Grünwald-Letnikov and backward Euler methods. First-order error estimate respect time rigorousl...

The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time–space dependent variable coefficients is challenging, especially when classical weak initial singularities are taken into account. In this paper, we introduce a concise technique to construct efficient time-stepping schemes step sizes for two-dimensional sub-diffusion and...

A class of explicit multistage time-stepping schemes with centered spatial di erencing and multigrid is considered for the compressible Euler and Navier-Stokes equations. These schemes are the basis for a family of computer programs ( ow codes with multigrid (FLOMG) series) currently used to solve a wide range of uid dynamics problems, including internal and external ows. In this paper, the com...

Atmospheric flows are characterized by a large range of length scales as well as strong gradients. The accurate simulation of such flows requires numerical algorithms with high spectral resolution, as well as the ability to provide nonoscillatory solutions across regions of high gradients. These flows exhibit a large range of time scales as well—the slowest waves propagate at the flow velocity ...

The flux-corrected-transport paradigm is generalized to finite element schemes based on arbitrary time-stepping. A conservative flux decomposition procedure is proposed for both convective and diffusive terms. Mathematical properties of positivity-preserving schemes are reviewed. A nonoscillatory low-order method is constructed by elimination of negative off-diagonal entries of the discrete tra...

Errors due to hardware or low level software problems, if detected, can be fixed by various schemes, such as recomputation from a checkpoint. Silent errors are errors in application state that have escaped low-level error detection. At extreme scale, where machines can perform astronomically many operations per second, silent errors threaten the validity of computed results. We propose a new pa...