536,617 research outputs found
Preconditioned iterative solution of the 2D Helmholtz equation
Using a finite element method to solve the Helmholtz equation leads to a sparse system of equations which in three dimensions is too large to solve directly. It is also non-Hermitian and highly indefinite and consequently difficult to solve iteratively. The approach taken in this paper is to precondition this linear system with a new preconditioner and then solve it iteratively using a Krylov subspace method. Numerical analysis shows the preconditioner to be effective on a simple 1D test problem, and results are presented showing considerable convergence acceleration for a number of different Krylov methods for more complex problems in 2D, as well as for the more general problem of harmonic disturbances to a non-stagnant steady flow
On stochastic differential equations driven by the renormalized square of the Gaussian white noise
We investigate the properties of the Wick square of Gaussian white noises
through a new method to perform non linear operations on Hida distributions.
This method lays in between the Wick product interpretation and the usual
definition of nonlinear functions. We prove on Ito-type formula and solve
stochastic differential equations driven by the renormalized square of the
Gaussian white noise. Our approach works with standard assumptions on the
coefficients of the equations, Lipschitz continuity and linear growth
condition, and produces existence and uniqueness results in the space where the
noise lives. The linear case is studied in details and positivity of the
solution is proved.Comment: 23 page
A Maximum Entropy Method for Solving the Boundary Value Problem of Second Order Ordinary Differential Equations
We propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations, which has a non-negative solution. The method applies the maximum entropy principle to approximating the solution numerically. The theoretical analysis and numerical examples show that our method is convergent
Contravariant Boussinesq equations for the simulation of wave transformation, breaking and run-up
We propose an integral form of the fully non-linear Boussinesq equations in
contravariant formulation, in which Christoffel symbols are avoided, in order to
simulate wave transformation phenomena, wave breaking and near shore
currents in computational domains representing the complex morphology of real
coastal regions. The motion equations retain the term related to the
approximation to the second order of the vertical vorticity. A new Upwind
Weighted Essentially Non-Oscillatory scheme for the solution of the fully non-
linear Boussinesq equations on generalised curvilinear coordinate systems is
proposed. The equations are rearranged in order to solve them by a high
resolution hybrid finite volume–finite difference scheme. The conservative part
of the above-mentioned equations, consisting of the convective terms and the
terms related to the free surface elevation, is discretised by a high-order shock-
capturing finite volume scheme; dispersive terms and the term related to the
approximation to the second order of the vertical vorticity are discretised by a
cell-centred finite difference scheme. The shock-capturing method makes it
possible to intrinsically model the wave breaking, therefore no additional terms
are needed to take into account the breaking related energy dissipation in the surf
zone. The model is applied on a real case regarding the simulation of wave fields
and nearshore currents in the coastal region opposite Pescara harbour (Italy)
Double power series method for approximating cosmological perturbations
We introduce a double power series method for finding approximate analytical
solutions for systems of differential equations commonly found in cosmological
perturbation theory. The method was set out, in a non-cosmological context, by
Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases
where perturbations are on sub-horizon scales. The FSN method is essentially an
extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding
approximate analytical solutions for ordinary differential equations. The FSN
method we use is applicable well beyond perturbation theory to solve systems of
ordinary differential equations, linear in the derivatives, that also depend on
a small parameter, which here we take to be related to the inverse wave-number.
We use the FSN method to find new approximate oscillating solutions in linear
order cosmological perturbation theory for a flat radiation-matter universe.
Together with this model's well known growing and decaying M\'esz\'aros
solutions, these oscillating modes provide a complete set of sub-horizon
approximations for the metric potential, radiation and matter perturbations.
Comparison with numerical solutions of the perturbation equations shows that
our approximations can be made accurate to within a typical error of 1%, or
better. We also set out a heuristic method for error estimation. A Mathematica
notebook which implements the double power series method is made available
online.Comment: 22 pages, 10 figures, 2 tables. Mathematica notebook available from
Github at https://github.com/AndrewWren/Double-power-series.gi
Smooth finite strain plasticity with non-local pressure support
The aim of this work is to introduce an alternative framework to solve problems of finite strain elastoplasticity including anisotropy and kinematic hardening coupled with any isotropic hyperelastic law. After deriving the constitutive equations and inequalities without any of the customary simplifications, we arrive at a new general elasto-plastic system. We integrate the elasto-plastic algebraico-differential system and replace the loading–unloading condition by a Chen–Mangasarian smooth function to obtain a non-linear system solved by a trust region method. Despite being non-standard, this approach is advantageous, since quadratic convergence is always obtained by the non-linear solver and very large steps can be used with negligible effect in the results. Discretized equilibrium is, in contrast with traditional approaches, smooth and well behaved. In addition, since no return mapping algorithm is used, there is no need to use a predictor. The work follows our previous studies of element technology and highly non-linear visco-elasticity. From a general framework, with exact linearization, systematic particularization is made to prototype constitutive models shown as examples. Our element with non-local pressure support is used. Examples illustrating the generality of the method are presented with excellent results
A fourier pseudospectral method for some computational aeroacoustics problems
A Fourier pseudospectral time-domain method is applied to wave propagation problems pertinent to computational aeroacoustics. The original algorithm of the Fourier pseudospectral time-domain method works for periodical problems without the interaction with physical boundaries. In this paper we develop a slip wall boundary condition, combined with buffer zone technique to solve some non-periodical problems. For a linear sound propagation problem whose governing equations could be transferred to ordinary differential equations in pseudospectral space, a new algorithm only requiring time stepping is developed and tested. For other wave propagation problems, the original algorithm has to be employed, and the developed slip wall boundary condition still works. The accuracy of the presented numerical algorithm is validated by benchmark problems, and the efficiency is assessed by comparing with high-order finite difference methods. It is indicated that the Fourier pseudospectral time-domain method, time stepping method, slip wall and absorbing boundary conditions combine together to form a fully-fledged computational algorithm
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