36 research outputs found
Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the navier-stokes equations
Recommended from our members
Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the navier-stokes equations
In this paper we analyze a finite element method applied to a continuous
downscaling data assimilation algorithm for the numerical approximation of the
two and three dimensional Navier-Stokes equations corresponding to given
measurements on a coarse spatial scale. For representing the coarse mesh
measurements we consider different types of interpolation operators including a
Lagrange interpolant. We obtain uniform-in-time estimates for the error between
a finite element approximation and the reference solution corresponding to the
coarse mesh measurements. We consider both the case of a plain Galerkin method
and a Galerkin method with grad-div stabilization. For the stabilized method we
prove error bounds in which the constants do not depend on inverse powers of
the viscosity. Some numerical experiments illustrate the theoretical results
Stabilization of Galerkin Finite Element Approximations to Transient Convection-Diffusion Problems
Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations
For classes of symplectic and symmetric time-stepping methods- trigonometric integrators and the Stormer-Verlet or leapfrog method-applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time
Adiabatic Integrators for Highly Oscillatory Second-Order Linear Differential Equations with Time-Varying Eigendecomposition
Overcoming instabilities in Verlet-I/r-RESPA with the mollified impulse method
Abstract. The primary objective of this paper is to explain the derivation of symplectic mollified Verlet-I/r-RESPA (MOLLY) methods that overcome linear and nonlinear instabilities that arise as numerical artifacts in Verlet-I/r-RESPA. These methods allow for lengthening of the longest time step used in molecular dynamics (MD). We provide evidence that MOLLY methods can take a longest time step that is 50 % greater than that of Verlet-I/r-RESPA, for a given drift, including no drift. A 350 % increase in the timestep is possible using MOLLY with mild Langevin damping while still computing dynamic properties accurately. Furthermore, longer time steps also enhance the scalability of multiple time stepping integrators that use the popular Particle Mesh Ewald method for computing full electrostatics, since the parallel bottleneck of the fast Fourier transform associated with PME is invoked less often. An additional objective of this paper is to give sufficient implementation details for these mollified integrators, so that interested users may implement them into their MD codes, or use the program ProtoMol in which we have implemented these methods