235 research outputs found
Destruction of the family of steady states in the planar problem of Darcy convection
The natural convection of incompressible fluid in a porous medium causes for
some boundary conditions a strong non-uniqueness in the form of a continuous
family of steady states. We are interested in the situation when these boundary
conditions are violated. The resulting destruction of the family of steady
states is studied via computer experiments based on a mimetic finite-difference
approach. Convection in a rectangular enclosure is considered under different
perturbations of boundary conditions (heat sources, infiltration). Two scenario
of the family of equilibria are found: the transformation to a limit cycle and
the formation of isolated convective patterns.Comment: 12 pages, 6 figure
Poisson integrators
An overview of Hamiltonian systems with noncanonical Poisson structures is
given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are
presented. Numerical integrators using generating functions, Hamiltonian
splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems
and Hamiltonian systems with a general Poisson structure. Nambu-Poisson systems
and the discrete gradient methods are also presented.Comment: 30 page
Staggered grids discretization in three-dimensional Darcy convection
We consider three-dimensional convection of an incompressible fluid saturated
in a parallelepiped with a porous medium. A mimetic finite-difference scheme
for the Darcy convection problem in the primitive variables is developed. It
consists of staggered nonuniform grids with five types of nodes, differencing
and averaging operators on a two-nodes stencil. The nonlinear terms are
approximated using special schemes. Two problems with different boundary
conditions are considered to study scenarios of instability of the state of
rest. Branching off of a continuous family of steady states was detected for
the problem with zero heat fluxes on two opposite lateral planes.Comment: 20 pages, 9 figure
Energy preserving model order reduction of the nonlinear Schr\"odinger equation
An energy preserving reduced order model is developed for two dimensional
nonlinear Schr\"odinger equation (NLSE) with plane wave solutions and with an
external potential. The NLSE is discretized in space by the symmetric interior
penalty discontinuous Galerkin (SIPG) method. The resulting system of
Hamiltonian ordinary differential equations are integrated in time by the
energy preserving average vector field (AVF) method. The mass and energy
preserving reduced order model (ROM) is constructed by proper orthogonal
decomposition (POD) Galerkin projection. The nonlinearities are computed for
the ROM efficiently by discrete empirical interpolation method (DEIM) and
dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and
mass are shown for the full order model (FOM) and for the ROM which ensures the
long term stability of the solutions. Numerical simulations illustrate the
preservation of the energy and mass in the reduced order model for the two
dimensional NLSE with and without the external potential. The POD-DMD makes a
remarkable improvement in computational speed-up over the POD-DEIM. Both
methods approximate accurately the FOM, whereas POD-DEIM is more accurate than
the POD-DMD
Poisson integrators for Volterra lattice equations
The Volterra lattice equations are completely integrable and possess
bi-Hamiltonian structure. They are integrated using partitioned Lobatto IIIA-B
methods which preserve the Poisson structure. Modified equations are derived
for the symplectic Euler and second order Lobatto IIIA-B method. Numerical
results confirm preservation of the corresponding Hamiltonians, Casimirs,
quadratic and cubic integrals in the long-term with different orders of
accuracy.Comment: 9 pages, 2 figure
Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation
In this paper, we compare three model order reduction methods: the proper
orthogonal decomposition (POD), discrete empirical interpolation method (DEIM)
and dynamic mode decomposition (DMD) for the optimal control of the convective
FitzHugh-Nagumo (FHN) equations. The convective FHN equations consists of the
semi-linear activator and the linear inhibitor equations, modeling blood
coagulation in moving excitable media. The semilinear activator equation leads
to a non-convex optimal control problem (OCP). The most commonly used method in
reduced optimal control is POD. We use DEIM and DMD to approximate efficiently
the nonlinear terms in reduced order models. We compare the accuracy and
computational times of three reduced-order optimal control solutions with the
full order discontinuous Galerkin finite element solution of the convection
dominated FHN equations with terminal controls. Numerical results show that POD
is the most accurate whereas POD-DMD is the fastest
Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements
This paper deals with pricing of European and American options, when the
underlying asset price follows Heston model, via the interior penalty
discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM
space discretization with Rannacher smoothing as time integrator with nonsmooth
initial and boundary conditions are illustrated for European vanilla options,
digital call and American put options. The convection dominated Heston model
for vanishing volatility is efficiently solved utilizing the adaptive dGFEM.
For fast solution of the linear complementary problem of the American options,
a projected successive over relaxation (PSOR) method is developed with the norm
preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option
pricing by conducting comparison analysis with other methods and numerical
experiments
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