4,497 research outputs found
On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations
The two-dimensional unsteady coupled Burgers' equations with moderate to
severe gradients, are solved numerically using higher-order accurate finite
difference schemes; namely the fourth-order accurate compact ADI scheme, and
the fourth-order accurate Du Fort Frankel scheme. The question of numerical
stability and convergence are presented. Comparisons are made between the
present schemes in terms of accuracy and computational efficiency for solving
problems with severe internal and boundary gradients. The present study shows
that the fourth-order compact ADI scheme is stable and efficient
A 8-neighbor model lattice Boltzmann method applied to mathematical-physical equations
© 2016. This version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A lattice Boltzmann method (LBM) 9-bit model is presented to solve mathematical-physical equations, such as, Laplace equation, Poisson equation, Wave equation and Burgers equation. The 9-bit model has been verified by several test cases. Numerical simulations, including 1D and 2D cases, of each problem are shown respectively. Comparisons are made between numerical predictions and analytic solutions or available numerical results from previous researchers. It turned out that the 9-bit model is computationally effective and accurate for all different mathematical-physical equations studied. The main benefits of the new model proposed is that it is faster than the previous existing models and has a better accuracy.Peer ReviewedPostprint (author's final draft
Proper orthogonal decomposition closure models for fluid flows: Burgers equation
This paper puts forth several closure models for the proper orthogonal
decomposition (POD) reduced order modeling of fluid flows. These new closure
models, together with other standard closure models, are investigated in the
numerical simulation of the Burgers equation. This simplified setting
represents just the first step in the investigation of the new closure models.
It allows a thorough assessment of the performance of the new models, including
a parameter sensitivity study. Two challenging test problems displaying moving
shock waves are chosen in the numerical investigation. The closure models and a
standard Galerkin POD reduced order model are benchmarked against the fine
resolution numerical simulation. Both numerical accuracy and computational
efficiency are used to assess the performance of the models
A machine learning framework for data driven acceleration of computations of differential equations
We propose a machine learning framework to accelerate numerical computations
of time-dependent ODEs and PDEs. Our method is based on recasting
(generalizations of) existing numerical methods as artificial neural networks,
with a set of trainable parameters. These parameters are determined in an
offline training process by (approximately) minimizing suitable (possibly
non-convex) loss functions by (stochastic) gradient descent methods. The
proposed algorithm is designed to be always consistent with the underlying
differential equation. Numerical experiments involving both linear and
non-linear ODE and PDE model problems demonstrate a significant gain in
computational efficiency over standard numerical methods
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
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