1,581 research outputs found
Line Integral Solution of Hamiltonian Systems with Holonomic Constraints
In this paper, we propose a second-order energy-conserving approximation
procedure for Hamiltonian systems with holonomic constraints. The derivation of
the procedure relies on the use of the so-called line integral framework. We
provide numerical experiments to illustrate theoretical findings.Comment: 30 pages, 3 figures, 4 table
Numerical stability of coupled differential equation with piecewise constant arguments
This paper deals with the stability of numerical solutions for a coupled differential equation with piecewise constant arguments. A sufficient condition such that the system is asymptotically stable is derived. Furthermore, when the linear θ-method is applied to this system, it is shown that the linear θ-method is asymptotically stable if and only if 1/2<θ≤1. Finally, some numerical experiments are given
A New Discontinuous Galerkin Finite Element Method for Directly Solving the Hamilton-Jacobi Equations
In this paper, we improve upon the discontinuous Galerkin (DG) method for
Hamilton-Jacobi (HJ) equation with convex Hamiltonians in (Y. Cheng and C.-W.
Shu, J. Comput. Phys. 223:398-415,2007) and develop a new DG method for
directly solving the general HJ equations. The new method avoids the
reconstruction of the solution across elements by utilizing the Roe speed at
the cell interface. Besides, we propose an entropy fix by adding penalty terms
proportional to the jump of the normal derivative of the numerical solution.
The particular form of the entropy fix was inspired by the Harten and Hyman's
entropy fix (A. Harten and J. M. Hyman. J. Comput. Phys. 50(2):235-269, 1983)
for Roe scheme for the conservation laws. The resulting scheme is compact,
simple to implement even on unstructured meshes, and is demonstrated to work
for nonconvex Hamiltonians. Benchmark numerical experiments in one dimension
and two dimensions are provided to validate the performance of the method
Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods
In this paper we are concerned with the analysis of a class of geometric
integrators, at first devised in [14, 18], which can be regarded as an
energy-conserving variant of Gauss collocation methods. With these latter they
share the property of conserving quadratic first integrals but, in addition,
they also conserve the Hamiltonian function itself. We here reformulate the
methods in a more convenient way, and propose a more refined analysis than that
given in [18] also providing, as a by-product, a practical procedure for their
implementation. A thorough comparison with the original Gauss methods is
carried out by means of a few numerical tests solving Hamiltonian and Poisson
problems.Comment: 28 pages, 2 figures, 4 table
Numerical Methods -- Lecture Notes 2014-2015
In these notes some basic numerical methods will be described. The
following topics are addressed: 1. Nonlinear Equations, 2. Linear
Systems, 3. Polynomial Interpolation and Approximation, 4. Trigonometric
Interpolation with DFT and FFT, 5. Numerical Integration, 6. Initial
Value Problems for ODEs, 7. Stiff Initial Value Problems, 8. Two-Point
Boundary Value Problems
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