An optimally accurate discrete regularization for second order timestepping methods for Navier–Stokes equations

Abstract

We propose a new, optimally accurate numerical regularization/stabilization for (a family of) second order timestepping methods for the Navier-Stokes equations (NSE). The method combines a linear treatment of the advection term, together with a stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure. We rigorously prove that the entire new family of methods are unconditionally stable and O(∆t 2) accurate. The idea of 'curvature stabilization' is new to CFD and is intended as an improvement over the commonly used 'speed stabilization', which is only first order accurate in time and can have an adverse affect on important flow quantities such as drag coefficients. Numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows.

Analysis of the Thermal Insulation of Textiles Using Thermography and CFD Simulation Based on Micro-CT Models

“…The Solidworks Flow Simulation 2014 software was used to model the heat transfer with the fi nite volume method. The applied software solves the energy conservation equations and the Navier-Stokes formulas [23,24], which describe the fl uid fl ow. This process enables one to estimate the simultaneous heat transfer in solid and fl uid media and incorporates the energy exchange between these media.…”

Analysis of Second Order Time Filtered Backward Euler Method for MHD Equations

“…In general since G-stability implies A-stability, the use of G-matrix is very common in BDF2 analysis, see e.g., [15] and references therein. These norms and properties are already have been given in [19]. With respect to notation of [19],(see page 392), analysis of the described method here uses the choices of θ = 1 and ν = 2 ,…”

“…To improve this behavior, time filters are used to stabilize the backward Euler discretizations in [14] for the classical numerical ODE theory.The present work extends the method of [7] tailored to MHD flows for constant time step. As it is mentioned in this study, the constant time step method is equivalent to a general second order, two step and A-stable method given in [9] and [19]. The scheme we consider is the time filtered backward Euler method, which is efficient, O(∆t 2 ) and amenable to implementation in existing legacy codes.…”

A gPAV-based unconditionally energy-stable scheme for incompressible flows with outflow/open boundaries

“…The unconditionally energy-stable schemes (see e.g. [45,48,51,25,35,18,43,30,8], among others) typically treat the nonlinear term in a fully implicit or linearized fashion, and can alleviate or eliminate the constraint on the time step size encountered with semi-implicit schemes. The main drawback of energy-stable schemes lies in that they typically require the solution of a system of nonlinear algebraic equations or a system of linear algebraic equations with a variable and time-dependent coefficient matrix within a time step [18].…”