13,022 research outputs found
High-order full discretization for anisotropic wave equations
Two-dimensional linear wave equation in anisotropic media, on a rectangular domain with initial conditions and periodic boundary conditions, is considered. The energy of the problem is contemplated. The space discretization is reached by means of finite differences on a uniform grid, paying attention to the mixed derivative of the equation. The discrete energy of the semi-discrete problem is introduced. For the time integration of the system of ordinary differential equations obtained, a fourth order exponential splitting method, which is a geometric integrator, is proposed. This time integrator is efficient and easy to implement. The stability condition for time step and space step ratio is deduced. Numerical experiments displaying the good behavior in the long time integration and the efficiency of the numerical solution are provided.MTM2015-66837-P del Ministerio de Economía y Competitivida
Elastic Wave Eigenmode Solver for Acoustic Waveguides
A numerical solver for the elastic wave eigenmodes in acoustic waveguides of
inhomogeneous cross-section is presented. Operating under the assumptions of
linear, isotropic materials, it utilizes a finite-difference method on a
staggered grid to solve for the acoustic eigenmodes of the vector-field elastic
wave equation. Free, fixed, symmetry, and anti-symmetry boundary conditions are
implemented, enabling efficient simulation of acoustic structures with
geometrical symmetries and terminations. Perfectly matched layers are also
implemented, allowing for the simulation of radiative (leaky) modes. The method
is analogous to eigenmode solvers ubiquitously employed in electromagnetics to
find waveguide modes, and enables design of acoustic waveguides as well as
seamless integration with electromagnetic solvers for optomechanical device
design. The accuracy of the solver is demonstrated by calculating
eigenfrequencies and mode shapes for common acoustic modes in several simple
geometries and comparing the results to analytical solutions where available or
to numerical solvers based on more computationally expensive methods
Kinetic Solvers with Adaptive Mesh in Phase Space
An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for
solving multi-dimensional kinetic equations by the discrete velocity method. A
Cartesian mesh for both configuration (r) and velocity (v) spaces is produced
using a tree of trees data structure. The mesh in r-space is automatically
generated around embedded boundaries and dynamically adapted to local solution
properties. The mesh in v-space is created on-the-fly for each cell in r-space.
Mappings between neighboring v-space trees implemented for the advection
operator in configuration space. We have developed new algorithms for solving
the full Boltzmann and linear Boltzmann equations with AMPS. Several recent
innovations were used to calculate the discrete Boltzmann collision integral
with dynamically adaptive mesh in velocity space: importance sampling,
multi-point projection method, and the variance reduction method. We have
developed an efficient algorithm for calculating the linear Boltzmann collision
integral for elastic and inelastic collisions in a Lorentz gas. New AMPS
technique has been demonstrated for simulations of hypersonic rarefied gas
flows, ion and electron kinetics in weakly ionized plasma, radiation and light
particle transport through thin films, and electron streaming in
semiconductors. We have shown that AMPS allows minimizing the number of cells
in phase space to reduce computational cost and memory usage for solving
challenging kinetic problems
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