1,258 research outputs found
An approximate solution of the MHD Falkner-Skan flow by Hermite functions pseudospectral method
Based on a new approximation method, namely pseudospectral method, a solution
for the three order nonlinear ordinary differential laminar boundary layer
Falkner-Skan equation has been obtained on the semi-infinite domain. The
proposed approach is equipped by the orthogonal Hermite functions that have
perfect properties to achieve this goal. This method solves the problem on the
semi-infinite domain without truncating it to a finite domain and transforming
domain of the problem to a finite domain. In addition, this method reduces
solution of the problem to solution of a system of algebraic equations. We also
present the comparison of this work with numerical results and show that the
present method is applicable.Comment: 15 pages, 4 figures; Published online in the journal of
"Communications in Nonlinear Science and Numerical Simulation
Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach
In this paper, we consider Burgers' equation with uncertain boundary and
initial conditions. The polynomial chaos (PC) approach yields a hyperbolic
system of deterministic equations, which can be solved by several numerical
methods. Here, we apply the correction procedure via reconstruction (CPR) using
summation-by-parts operators. We focus especially on stability, which is proven
for CPR methods and the systems arising from the PC approach. Due to the usage
of split-forms, the major challenge is to construct entropy stable numerical
fluxes. For the first time, such numerical fluxes are constructed for all
systems resulting from the PC approach for Burgers' equation. In numerical
tests, we verify our results and show also the advantage of the given ansatz
using CPR methods. Moreover, one of the simulations, i.e. Burgers' equation
equipped with an initial shock, demonstrates quite fascinating observations.
The behaviour of the numerical solutions from several methods (finite volume,
finite difference, CPR) differ significantly from each other. Through careful
investigations, we conclude that the reason for this is the high sensitivity of
the system to varying dissipation. Furthermore, it should be stressed that the
system is not strictly hyperbolic with genuinely nonlinear or linearly
degenerate fields
Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
Elliptic partial differential equations with diffusion coefficients of
lognormal form, that is , where is a Gaussian random field, are
considered. We study the summability properties of the Hermite
polynomial expansion of the solution in terms of the countably many scalar
parameters appearing in a given representation of . These summability
results have direct consequences on the approximation rates of best -term
truncated Hermite expansions. Our results significantly improve on the state of
the art estimates available for this problem. In particular, they take into
account the support properties of the basis functions involved in the
representation of , in addition to the size of these functions. One
interesting conclusion from our analysis is that in certain relevant cases, the
Karhunen-Lo\`eve representation of may not be the best choice concerning
the resulting sparsity and approximability of the Hermite expansion
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