5 research outputs found
A high-order artificial compressibility method based on Taylor series time-stepping for variable density flow
In this paper, we introduce a fourth-order accurate finite element method for
incompressible variable density flow. The method is implicit in time and
constructed with the Taylor series technique, and uses standard high-order
Lagrange basis functions in space. Taylor series time-stepping relies on time
derivative correction terms to achieve high-order accuracy. We provide detailed
algorithms to approximate the time derivatives of the variable density
Navier-Stokes equations. Numerical validations confirm a fourth-order accuracy
for smooth problems. We also numerically illustrate that the Taylor series
method is unsuitable for problems where regularity is lost by solving the 2D
Rayleigh-Taylor instability problem
Variable Time Step Method of DAHLQUIST, LINIGER and NEVANLINNA (DLN) for a Corrected Smagorinsky Model
Turbulent flows strain resources, both memory and CPU speed. The DLN method
has greater accuracy and allows larger time steps, requiring less memory and
fewer FLOPS. The DLN method can also be implemented adaptively. The classical
Smagorinsky model, as an effective way to approximate a (resolved) mean
velocity, has recently been corrected to represent a flow of energy from
unresolved fluctuations to the (resolved) mean velocity. In this paper, we
apply a family of second-order, G-stable time-stepping methods proposed by
Dahlquist, Liniger, and Nevanlinna (the DLN method) to one corrected
Smagorinsky model and provide the detailed numerical analysis of the stability
and consistency. We prove that the numerical solutions under any arbitrary time
step sequences are unconditionally stable in the long term and converge at
second order. We also provide error estimate under certain time step condition.
Numerical tests are given to confirm the rate of convergence and also to show
that the adaptive DLN algorithm helps to control numerical dissipation so that
backscatter is visible