42 research outputs found
Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent convection-diffusion-reaction equations
This paper considers the numerical solution of time-dependent convection-diffusion-reaction equations. We shall employ combinations of streamline-upwind Petrov-Galerkin (SUPG) and local projection stabilization (LPS) methods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) methods and continuous Galerkin-Petrov (cGP) methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. Finally the long-time behavior of overshoots and undershoots is investigated
Numerical studies of higher order variational time stepping schemes for evolutionary Navier--Stokes equations
We present in this paper numerical studies of higher order variational time stepping schemes com-bined with finite element methods for simulations of the evolutionary Navier--Stokes equations. In particular, conforming inf-sup stable pairs of finite element spaces for approximating velocity and pressure are used as spatial discretization while continuous Galerkin--Petrov methods (cGP) and discontinuous Galerkin (dG) methods are applied as higher order variational time discretizations. Numerical results for the well-known problem of incompressible flows around a circle will be presented
Higher order continuous Galerkin--Petrov time stepping schemes for transient convection-diffusion-reaction equations
We present the analysis for the higher order continuous Galerkin--Petrov (cGP) time discretization schemes in combination with the one-level local projection stabilization in space applied to time-dependent convection-diffusion-reaction problems. Optimal a-priori error estimates will be proved. Numerical studies support the theoretical results. Furthermore, a numerical comparison between continuous Galerkin--Petrov and discontinuous Galerkin time discretization schemes will be given
Recommended from our members
Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent convection-diffusion-reaction equations
This paper considers the numerical solution of time-dependent
convection-diffusion-reaction equations. We shall employ combinations of
streamline-upwind Petrov-Galerkin (SUPG) and local projection stabilization
(LPS) methods in space with the higher order variational time discretization
schemes. In particular, we consider time discretizations by discontinuous
Galerkin (dG) methods and continuous Galerkin-Petrov (cGP) methods. Several
numerical tests have been performed to assess the accuracy of combinations of
spatial and temporal discretization schemes. Furthermore, the dependence of
the results on the stabilization parameters of the spatial discretizations
are discussed. Finally the long-time behavior of overshoots and undershoots
is investigated
An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics
We present a three-point iterative method without memory for solving
nonlinear equations in one variable. The proposed method provides convergence
order eight with four function evaluations per iteration. Hence, it possesses a
very high computational efficiency and supports Kung and Traub's conjecture.
The construction, the convergence analysis, and the numerical implementation of
the method will be presented. Using several test problems, the proposed method
will be compared with existing methods of convergence order eight concerning
accuracy and basin of attraction. Furthermore, some measures are used to judge
methods with respect to their performance in finding the basin of attraction.Comment: arXiv admin note: substantial text overlap with arXiv:1508.0174
Recommended from our members
Non-nested multi-grid solvers for mixed divergence-free Scott-Vogelius discretizations
Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure
Recommended from our members
Robust arbitrary order mixed finite element methods for the incompressible Stokes equations
Standard mixed finite element methods for the incompressible
Navier-Stokes equations that relax the divergence constraint are not robust
against large irrotational forces in the momentum balance and the velocity
error depends on the continuous pressure. This robustness issue can be
completely cured by using divergence-free mixed finite elements which deliver
pressure-independent velocity error estimates. However, the construction of
H1-conforming, divergence-free mixed finite element methods is rather
difficult. Instead, we present a novel approach for the construction of
arbitrary order mixed finite element methods which deliver
pressure-independent velocity errors. The approach does not change the trial
functions but replaces discretely divergence-free test functions in some
operators of the weak formulation by divergence-free ones. This modification
is applied to inf-sup stable conforming and nonconforming mixed finite
element methods of arbitrary order in two and three dimensions. Optimal
estimates for the incompressible Stokes equations are proved for the H1 and
L2 errors of the velocity and the L2 error of the pressure. Moreover, both
velocity errors are pressure-independent, demonstrating the improved
robustness. Several numerical examples illustrate the results
Recommended from our members
Higher-order discontinuous Galerkin time stepping and local projection stabilization techniques for the transient Stokes problem
We introduce and analyze discontinuous Galerkin time discretizations
coupled with continuous finite element methods based on equal-order
interpolation in space for velocity and pressure in transient Stokes
problems. Spatial stability of the pressure is ensured by adding a
stabilization term based on local projection. We present error estimates for
the semi-discrete problem after discretization in space only and for the
fully discrete problem. The fully discrete pressure shows an instability in
the limit of small time step length. Numerical tests are presented which
confirm our theoretical results including the pressure instability