14,894 research outputs found
Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems
This paper studies the spatial manifestations of order reduction that occur
when time-stepping initial-boundary-value problems (IBVPs) with high-order
Runge-Kutta methods. For such IBVPs, geometric structures arise that do not
have an analog in ODE IVPs: boundary layers appear, induced by a mismatch
between the approximation error in the interior and at the boundaries. To
understand those boundary layers, an analysis of the modes of the numerical
scheme is conducted, which explains under which circumstances boundary layers
persist over many time steps. Based on this, two remedies to order reduction
are studied: first, a new condition on the Butcher tableau, called weak stage
order, that is compatible with diagonally implicit Runge-Kutta schemes; and
second, the impact of modified boundary conditions on the boundary layer theory
is analyzed.Comment: 41 pages, 9 figure
Constraint-consistent Runge-Kutta methods for one-dimensional incompressible multiphase flow
New time integration methods are proposed for simulating incompressible
multiphase flow in pipelines described by the one-dimensional two-fluid model.
The methodology is based on 'half-explicit' Runge-Kutta methods, being explicit
for the mass and momentum equations and implicit for the volume constraint.
These half-explicit methods are constraint-consistent, i.e., they satisfy the
hidden constraints of the two-fluid model, namely the volumetric flow
(incompressibility) constraint and the Poisson equation for the pressure. A
novel analysis shows that these hidden constraints are present in the
continuous, semi-discrete, and fully discrete equations.
Next to constraint-consistency, the new methods are conservative: the
original mass and momentum equations are solved, and the proper shock
conditions are satisfied; efficient: the implicit constraint is rewritten into
a pressure Poisson equation, and the time step for the explicit part is
restricted by a CFL condition based on the convective wave speeds; and
accurate: achieving high order temporal accuracy for all solution components
(masses, velocities, and pressure). High-order accuracy is obtained by
constructing a new third order Runge-Kutta method that satisfies the additional
order conditions arising from the presence of the constraint in combination
with time-dependent boundary conditions.
Two test cases (Kelvin-Helmholtz instabilities in a pipeline and liquid
sloshing in a cylindrical tank) show that for time-independent boundary
conditions the half-explicit formulation with a classic fourth-order
Runge-Kutta method accurately integrates the two-fluid model equations in time
while preserving all constraints. A third test case (ramp-up of gas production
in a multiphase pipeline) shows that our new third order method is preferred
for cases featuring time-dependent boundary conditions
On the convergence of Lawson methods for semilinear stiff problems
Since their introduction in 1967, Lawson methods have achieved constant
interest in the time discretization of evolution equations. The methods were
originally devised for the numerical solution of stiff differential equations.
Meanwhile, they constitute a well-established class of exponential integrators.
The popularity of Lawson methods is in some contrast to the fact that they may
have a bad convergence behaviour, since they do not satisfy any of the stiff
order conditions. The aim of this paper is to explain this discrepancy. It is
shown that non-stiff order conditions together with appropriate regularity
assumptions imply high-order convergence of Lawson methods. Note, however, that
the term regularity here includes the behaviour of the solution at the
boundary. For instance, Lawson methods will behave well in the case of periodic
boundary conditions, but they will show a dramatic order reduction for, e.g.,
Dirichlet boundary conditions. The precise regularity assumptions required for
high-order convergence are worked out in this paper and related to the
corresponding assumptions for splitting schemes. In contrast to previous work,
the analysis is based on expansions of the exact and the numerical solution
along the flow of the homogeneous problem. Numerical examples for the
Schr\"odinger equation are included
Spatially partitioned embedded Runge-Kutta Methods
We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory
Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems
Explicit Runge-Kutta schemes with large stable step sizes are developed for
integration of high order spectral difference spatial discretization on
quadrilateral grids. The new schemes permit an effective time step that is
substantially larger than the maximum admissible time step of standard explicit
Runge-Kutta schemes available in literature. Furthermore, they have a small
principal error norm and admit a low-storage implementation. The advantages of
the new schemes are demonstrated through application to the Euler equations and
the linearized Euler equations.Comment: 37 pages, 3 pages of appendi
HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II. Optimization of the Runge-Kutta smoother
Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge-Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge-Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space-time discontinuous Galerkin finite element discretization of the advection-diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained
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