23,954 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
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
Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
This article serves as a summary outlining the mathematical entropy analysis
of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD
equations as they are particularly useful for mathematically modeling a wide
variety of magnetized fluids. In order to be self-contained we first motivate
the physical properties of a magnetic fluid and how it should behave under the
laws of thermodynamics. Next, we introduce a mathematical model built from
hyperbolic partial differential equations (PDEs) that translate physical laws
into mathematical equations. After an overview of the continuous analysis, we
thoroughly describe the derivation of a numerical approximation of the ideal
MHD system that remains consistent to the continuous thermodynamic principles.
The derivation of the method and the theorems contained within serve as the
bulk of the review article. We demonstrate that the derived numerical
approximation retains the correct entropic properties of the continuous model
and show its applicability to a variety of standard numerical test cases for
MHD schemes. We close with our conclusions and a brief discussion on future
work in the area of entropy consistent numerical methods and the modeling of
plasmas
First-Order System Least Squares and the Energetic Variational Approach for Two-Phase Flow
This paper develops a first-order system least-squares (FOSLS) formulation
for equations of two-phase flow. The main goal is to show that this
discretization, along with numerical techniques such as nested iteration,
algebraic multigrid, and adaptive local refinement, can be used to solve these
types of complex fluid flow problems. In addition, from an energetic
variational approach, it can be shown that an important quantity to preserve in
a given simulation is the energy law. We discuss the energy law and inherent
structure for two-phase flow using the Allen-Cahn interface model and indicate
how it is related to other complex fluid models, such as magnetohydrodynamics.
Finally, we show that, using the FOSLS framework, one can still satisfy the
appropriate energy law globally while using well-known numerical techniques.Comment: 22 pages, 8 figures submitted to Journal of Computational Physic
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