10,188 research outputs found
Adaptive Energy Preserving Methods for Partial Differential Equations
A method for constructing first integral preserving numerical schemes for
time-dependent partial differential equations on non-uniform grids is
presented. The method can be used with both finite difference and partition of
unity approaches, thereby also including finite element approaches. The schemes
are then extended to accommodate -, - and -adaptivity. The method is
applied to the Korteweg-de Vries equation and the Sine-Gordon equation and
results from numerical experiments are presented.Comment: 27 pages; some changes to notation and figure
Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling
In the present article we describe a few simple and efficient finite volume
type schemes on moving grids in one spatial dimension combined with appropriate
predictor-corrector method to achieve higher resolution. The underlying finite
volume scheme is conservative and it is accurate up to the second order in
space. The main novelty consists in the motion of the grid. This new dynamic
aspect can be used to resolve better the areas with large solution gradients or
any other special features. No interpolation procedure is employed, thus
unnecessary solution smearing is avoided, and therefore, our method enjoys
excellent conservation properties. The resulting grid is completely
redistributed according the choice of the so-called monitor function. Several
more or less universal choices of the monitor function are provided. Finally,
the performance of the proposed algorithm is illustrated on several examples
stemming from the simple linear advection to the simulation of complex shallow
water waves. The exact well-balanced property is proven. We believe that the
techniques described in our paper can be beneficially used to model tsunami
wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to
Geosciences. Other author's papers can be downloaded at
http://www.denys-dutykh.com
A moving mesh method with variable relaxation time
We propose a moving mesh adaptive approach for solving time-dependent partial
differential equations. The motion of spatial grid points is governed by a
moving mesh PDE (MMPDE) in which a mesh relaxation time \tau is employed as a
regularization parameter. Previously reported results on MMPDEs have invariably
employed a constant value of the parameter \tau. We extend this standard
approach by incorporating a variable relaxation time that is calculated
adaptively alongside the solution in order to regularize the mesh appropriately
throughout a computation. We focus on singular problems involving self-similar
blow-up to demonstrate the advantages of using a variable relaxation ime over a
fixed one in terms of accuracy, stability and efficiency.Comment: 21 page
Moving mesh finite difference solution of non-equilibrium radiation diffusion equations
A moving mesh finite difference method based on the moving mesh partial
differential equation is proposed for the numerical solution of the 2T model
for multi-material, non-equilibrium radiation diffusion equations. The model
involves nonlinear diffusion coefficients and its solutions stay positive for
all time when they are positive initially. Nonlinear diffusion and preservation
of solution positivity pose challenges in the numerical solution of the model.
A coefficient-freezing predictor-corrector method is used for nonlinear
diffusion while a cutoff strategy with a positive threshold is used to keep the
solutions positive. Furthermore, a two-level moving mesh strategy and a sparse
matrix solver are used to improve the efficiency of the computation. Numerical
results for a selection of examples of multi-material non-equilibrium radiation
diffusion show that the method is capable of capturing the profiles and local
structures of Marshak waves with adequate mesh concentration. The obtained
numerical solutions are in good agreement with those in the existing
literature. Comparison studies are also made between uniform and adaptive
moving meshes and between one-level and two-level moving meshes.Comment: 29 page
R-adaptive multisymplectic and variational integrators
Moving mesh methods (also called r-adaptive methods) are space-adaptive
strategies used for the numerical simulation of time-dependent partial
differential equations. These methods keep the total number of mesh points
fixed during the simulation, but redistribute them over time to follow the
areas where a higher mesh point density is required. There are a very limited
number of moving mesh methods designed for solving field-theoretic partial
differential equations, and the numerical analysis of the resulting schemes is
challenging. In this paper we present two ways to construct r-adaptive
variational and multisymplectic integrators for (1+1)-dimensional Lagrangian
field theories. The first method uses a variational discretization of the
physical equations and the mesh equations are then coupled in a way typical of
the existing r-adaptive schemes. The second method treats the mesh points as
pseudo-particles and incorporates their dynamics directly into the variational
principle. A user-specified adaptation strategy is then enforced through
Lagrange multipliers as a constraint on the dynamics of both the physical field
and the mesh points. We discuss the advantages and limitations of our methods.
Numerical results for the Sine-Gordon equation are also presented.Comment: 65 pages, 13 figure
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