A perfectly matched layer approach to the linearized shallow water equations models

Monthly Weather Review, 132 No.6, (2004), 1369 – 1378.A limited-area model of linearized shallow water equations (SWE) on an f-plane for a rectangular domain is considered. The rectangular domain is extended to include the so-called perfectly matched layer (PML) as an absorbing boundary condition. Following the proponent of the original method, the equations are obtained in this layer by splitting the shallow water equations in the coordinate directions and introducing the absorption coefficients. The performance of the PML as an absorbing boundary treatment is demonstrated using a commonly employed bell-shaped Gaussian initially introduced at the center of the rectangular physical domain

A POD reduced order model for resolving angular direction in neutron/photon transport problems

publisher: Elsevier articletitle: A POD reduced order model for resolving angular direction in neutron/photon transport problems journaltitle: Journal of Computational Physics articlelink: http://dx.doi.org/10.1016/j.jcp.2015.04.043 content_type: article copyright: Copyright © 2015 Elsevier Inc. All rights reserved.publisher: Elsevier articletitle: A POD reduced order model for resolving angular direction in neutron/photon transport problems journaltitle: Journal of Computational Physics articlelink: http://dx.doi.org/10.1016/j.jcp.2015.04.043 content_type: article copyright: Copyright © 2015 Elsevier Inc. All rights reserved.publisher: Elsevier articletitle: A POD reduced order model for resolving angular direction in neutron/photon transport problems journaltitle: Journal of Computational Physics articlelink: http://dx.doi.org/10.1016/j.jcp.2015.04.043 content_type: article copyright: Copyright © 2015 Elsevier Inc. All rights reserved

Non-linear model reduction for the Navier–Stokes equations using residual DEIM method

This article presents a new reduced order model based upon proper orthogonal decomposition (POD) for solving the Navier–Stokes equations. The novelty of the method lies in its treatment of the equation's non-linear operator, for which a new method is proposed that provides accurate simulations within an efficient framework. The method itself is a hybrid of two existing approaches, namely the quadratic expansion method and the Discrete Empirical Interpolation Method (DEIM), that have already been developed to treat non-linear operators within reduced order models. The method proposed applies the quadratic expansion to provide a first approximation of the non-linear operator, and DEIM is then used as a corrector to improve its representation. In addition to the treatment of the non-linear operator the POD model is stabilized using a Petrov–Galerkin method. This adds artificial dissipation to the solution of the reduced order model which is necessary to avoid spurious oscillations and unstable solutions.A demonstration of the capabilities of this new approach is provided by solving the incompressible Navier–Stokes equations for simulating a flow past a cylinder and gyre problems. Comparisons are made with other treatments of non-linear operators, and these show the new method to provide significant improvements in the solution's accuracy

A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting

This work presents the first application of a non-intrusive reduced order method to model solid interacting with compressible fluid flows to simulate crack initiation and propagation. In the high fidelity model, the coupling process is achieved by introducing a source term into the momentum equation, which represents the effects of forces of the solid on the fluid. A combined single and smeared crack model with the Mohr–Coulomb failure criterion is used to simulate crack initiation and propagation. The non-intrusive reduced order method is then applied to compressible fluid and fractured solid coupled modelling where the computational cost involved in the full high fidelity simulation is high. The non-intrusive reduced order model (NIROM) developed here is constructed through proper orthogonal decomposition (POD) and a radial basis function (RBF) multi-dimensional interpolation method.The performance of the NIROM for solid interacting with compressible fluid flows, in the presence of fracture models, is illustrated by two complex test cases: an immersed wall in a fluid and a blasting test case. The numerical simulation results show that the NIROM is capable of capturing the details of compressible fluids and fractured solids while the CPU time is reduced by several orders of magnitude. In addition, the issue of whether or not to subtract the mean from the snapshots before applying POD is discussed in this paper. It is shown that solutions of the NIROM, without mean subtracted before constructing the POD basis, captured more details than the NIROM with mean subtracted from snapshots

Analysis of the Turkel-Zwas Scheme for the Shallow-Water Equations

J. Computational Physics, 81, (1989), 277–299.A transfer function analysis is used to analyze the Turkel-Zwas explicit large time step scheme applied to the shallow-water equations. The transfer function concept leads to insight into the behavior of this discretization scheme in terms of comparison between continuous and discrete amplitude, phase, and group velocity coefficients. The dependence of the distortion increases with the increase in the time-step size taken for the Turkel-Zwas scheme, which depends on the ratio between a coarse and a tine mesh. A comparison with earlier results of Schoenstadt (Naval Post-Graduate School Report NPS-53-79-001, 1978 (unpublished)) shows the Turkel-Zwas scheme to give reasonable results up to time steps three times larger than the CFL limit.Partially funded by the U.S. Department of Energy Contract DE-FC05-85ER250000

A perfectly matched layer formulation for the nonlinear shallow water equations models: The split equation approach

A limited-area model of nonlinear shallow water equations (SWE) with the Coriolis term in a rectangular domain is considered. The rectangular domain is extended to include the so-called perfectly matched layer (PML). Following the proponent of the original method, the PML equations are obtained by splitting the shallow water equations in the coordinate directions and introducing the damping terms. The efficacy of the PML boundary treatment is demonstrated in the case where a Gaussian pulse is initially introduced at the center of the rectangular physical domain. A systematic study is carried out for different mean convection speeds, and various values of the PML width and the damping coefficients. For the purpose of comparison, a reference solution is obtained on a fine grid on the extended domain with the characteristic boundary conditions. The L2 difference in the height field between the solution with the PML boundary treatment and the reference solution along a line at a downstream position in the interior domain is computed. The PML boundary treatment is found to yield better accuracy compared with both the characterisitic boundary conditions and the Engquist-Majda absorbing boundary conditions on an identical grid

Optimal control of flow with discontinuities

Optimal control of the 1-D Riemann problem of Euler equations whose solution is characterized by discontinuities is carried out by both nonsmooth and smooth op- timization methods. By matching a desired flow to the numerical model for a given time window we effectively change the location of discontinuities. The control pa- rameters are chosen to be the initial values for pressure and density fields. Existence of solutions for the optimal control problem is proven. A high resolution model and a model with artificial viscosity, implementing two different numerical methods, are used to solve the forward model. The cost functional is the weighted difference be- tween the numerical values and the observations for pressure, density and velocity. The observations are constructed from the analytical solution. We consider either distributed observations in time or observations calculated at the end of the assimi- lation window. We consider two different time horizons and two sets of observations. The gradient (respectively a subgradient) of the cost functional, obtained from the adjoint of the discrete forward model, are employed for the smooth minimization (respectively for the nonsmooth minimization) algorithm. Discontinuity detection improves the performance of the minimizer for the model with artificial viscosity by selecting the points where the shock occurs (and these points are then removed from Preprint submitted to Elsevier Science 26 March 2002 the cost functional and its gradient). The numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well the observations. The algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon

Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems I: Reaction-diffusion Type

AbstractWe consider the bilinear finite element method on a Shishkin mesh for the singularly perturbed elliptic boundary value problem −ϵ2(ι2uιx2 + ι2uιy2) + a(x,y)u = f(x,y) in two space dimensions. By using a very sophisticated asymptotic expansion of Han et al. [1] and the technique we used in [2], we prove that our method achieves almost second-order uniform convergence rate in L2-norm. Numerical results confirm our theoretical analysis