590 research outputs found
Wave dynamics on networks: method and application to the sine-Gordon equation
We consider a scalar Hamiltonian nonlinear wave equation formulated on
networks; this is a non standard problem because these domains are not locally
homeomorphic to any subset of the Euclidean space. More precisely, we assume
each edge to be a 1D uniform line with end points identified with graph
vertices. The interface conditions at these vertices are introduced and
justified using conservation laws and an homothetic argument. We present a
detailed methodology based on a symplectic finite difference scheme together
with a special treatment at the junctions to solve the problem and apply it to
the sine-Gordon equation. Numerical results on a simple graph containing four
loops show the performance of the scheme for kinks and breathers initial
conditions.Comment: 31 pages, 9 figures, 2 tables, 41 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
The meshless methods for numerical solution of the nonlinear Klein-Gordon equation
In this paper, we develop the numerical solution of nonlinear Klein-Gordon equation (NKGE) using the meshless methods. The finite difference scheme and the radial basis functions (RBFs) collocation methods are used to discretize time derivative and spatial derivatives, respectively. Numerical results are given to confirm the accuracy and efficiency of the presented schemes.Publisher's Versio
Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems
Reduced basis methods are popular for approximately solving large and complex
systems of differential equations. However, conventional reduced basis methods
do not generally preserve conservation laws and symmetries of the full order
model. Here, we present an approach for reduced model construction, that
preserves the symplectic symmetry of dissipative Hamiltonian systems. The
method constructs a closed reduced Hamiltonian system by coupling the full
model with a canonical heat bath. This allows the reduced system to be
integrated with a symplectic integrator, resulting in a correct dissipation of
energy, preservation of the total energy and, ultimately, in the stability of
the solution. Accuracy and stability of the method are illustrated through the
numerical simulation of the dissipative wave equation and a port-Hamiltonian
model of an electric circuit
Finite-difference schemes for nonlinear wave equation that inherit energy conservation property
AbstractWe propose two general finite-difference schemes that inherit energy conservation property from nonlinear wave equations, such as the nonlinear Klein–Gordon equation (NLKGE). One of proposed schemes is implicit and another is explicit. Many studies exist on FDSs that inherit energy conservation property from NLKGE and we can derive all of their schemes from the proposed general schemes in this paper. The most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts, which implies that the inherited properties are satisfied exactly. Because of this the derived schemes are expected to be numerically stable and yield solutions converging to PDE solutions. We make new FDSs for Fermi–Pasta–Ulam equation, string vibration equation, Shimoji–Kawai equation (SKE) and Ebihara equation and verify numerically the inheritance of the energy conservation property for NLKGE and SKE
Integrable Quantum Field Theories in Finite Volume: Excited State Energies
We develop a method of computing the excited state energies in Integrable
Quantum Field Theories (IQFT) in finite geometry, with spatial coordinate
compactified on a circle of circumference R. The IQFT ``commuting
transfer-matrices'' introduced by us (BLZ) for Conformal Field Theories (CFT)
are generalized to non-conformal IQFT obtained by perturbing CFT with the
operator . We study the models in which the fusion relations for
these ``transfer-matrices'' truncate and provide closed integral equations
which generalize the equations of Thermodynamic Bethe Ansatz to excited states.
The explicit calculations are done for the first excited state in the ``Scaling
Lee-Yang Model''.Comment: 54 pages, harvmac, epsf, TeX file and postscript figures packed in a
single selfextracting uufile. Compiles only in the `Big' mode with harvma
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