3,338 research outputs found
Essentially nonoscillatory postprocessing filtering methods
High order accurate centered flux approximations used in the computation of numerical solutions to nonlinear partial differential equations produce large oscillations in regions of sharp transitions. Here, we present a new class of filtering methods denoted by Essentially Nonoscillatory Least Squares (ENOLS), which constructs an upgraded filtered solution that is close to the physically correct weak solution of the original evolution equation. Our method relies on the evaluation of a least squares polynomial approximation to oscillatory data using a set of points which is determined via the ENO network. Numerical results are given in one and two space dimensions for both scalar and systems of hyperbolic conservation laws. Computational running time, efficiency, and robustness of method are illustrated in various examples such as Riemann initial data for both Burgers' and Euler's equations of gas dynamics. In all standard cases, the filtered solution appears to converge numerically to the correct solution of the original problem. Some interesting results based on nonstandard central difference schemes, which exactly preserve entropy, and have been recently shown generally not to be weakly convergent to a solution of the conservation law, are also obtained using our filters
Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems
Stochastic physical problems governed by nonlinear conservation laws are
challenging due to solution discontinuities in stochastic and physical space.
In this paper, we present a level set method to track discontinuities in
stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed
function that vanishes at discontinuities, the iso-zero of the level set
problem coincide with the discontinuities of the conservation law. The level
set problem is solved on a sequence of successively finer grids in stochastic
space. The method is adaptive in the sense that costly evaluations of the
conservation law of interest are only performed in the vicinity of the
discontinuities during the refinement stage. In regions of stochastic space
where the solution is smooth, a surrogate method replaces expensive evaluations
of the conservation law. The proposed method is tested in conjunction with
different sets of localized orthogonal basis functions on simplex elements, as
well as frames based on piecewise polynomials conforming to the level set
function. The performance of the proposed method is compared to existing
adaptive multi-element generalized polynomial chaos methods
A p-version finite element method for steady incompressible fluid flow and convective heat transfer
A new p-version finite element formulation for steady, incompressible fluid flow and convective heat transfer problems is presented. The steady-state residual equations are obtained by considering a limiting case of the least-squares formulation for the transient problem. The method circumvents the Babuska-Brezzi condition, permitting the use of equal-order interpolation for velocity and pressure, without requiring the use of arbitrary parameters. Numerical results are presented to demonstrate the accuracy and generality of the method
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