23 research outputs found
Fourth-order time-stepping for stiff PDEs on the sphere
We present in this paper algorithms for solving stiff PDEs on the unit sphere
with spectral accuracy in space and fourth-order accuracy in time. These are
based on a variant of the double Fourier sphere method in coefficient space
with multiplication matrices that differ from the usual ones, and
implicit-explicit time-stepping schemes. Operating in coefficient space with
these new matrices allows one to use a sparse direct solver, avoids the
coordinate singularity and maintains smoothness at the poles, while
implicit-explicit schemes circumvent severe restrictions on the time-steps due
to stiffness. A comparison is made against exponential integrators and it is
found that implicit-explicit schemes perform best. Implementations in MATLAB
and Chebfun make it possible to compute the solution of many PDEs to high
accuracy in a very convenient fashion
Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators
Dozens of exponential integration formulas have been proposed for the
high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries
and Ginzburg-Landau equations. We report the results of extensive comparisons
in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and
higher order methods, and periodic semilinear stiff PDEs with constant
coefficients. Our conclusion is that it is hard to do much better than one of
the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews
Computing planar and spherical choreographies
An algorithm is presented for numerical computation of choreographies in the
plane in a Newtonian potential and on the sphere in a cotangent potential. It
is based on stereographic projection, approximation by trigonometric
polynomials, and quasi-Newton and Newton optimization methods with exact
gradient and exact Hessian matrix. New choreographies on the sphere are
presented
The linear sampling method for random sources
We present an extension of the linear sampling method for solving the
sound-soft inverse acoustic scattering problem with randomly distributed point
sources. The theoretical justification of our sampling method is based on the
Helmholtz--Kirchhoff identity, the cross-correlation between measurements, and
the volume and imaginary near-field operators, which we introduce and analyze.
Implementations in MATLAB using boundary elements, the SVD, Tikhonov
regularization, and Morozov's discrepancy principle are also discussed. We
demonstrate the robustness and accuracy of our algorithms with several
numerical experiments in two dimensions
Gradient Span Analysis Method: Application to the Multipoint Aerodynamic Shape Optimization of a Turbine Cascade
This paper presents the application of the gradient span analysis (GSA) method to the multipoint optimization of the two-dimensional LS89 turbine distributor. The cost function (total pressure loss) and the constraint (mass flow rate) are computed from the resolution of the Reynolds-averaged Navier-Stokes equations. The penalty method is used to replace the constrained optimization problem with an unconstrained problem. The optimization process is steered by a gradient-based quasi-Newton algorithm. The gradient of the cost function with respect to design variables is obtained with the discrete adjoint method, which ensures an efficient computation time independent of the number of design variables. The GSA method gives a minimal set of operating conditions to insert into the weighted sum model to solve the multipoint optimization problem. The weights associated to these conditions are computed with the utopia point method. The single-point optimization at the nominal condition and the multipoint optimization over a wide range of conditions of the LS89 blade are compared. The comparison shows the strong advantages of the multipoint optimization with the GSA method and utopia-point weighting over the traditional single-point optimization
Extension of Chebfun to Periodic Functions
Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary differential equations. Differences from the nonperiodic Chebyshev case are highlighted