Efficient Low Dissipative High Order Schemes for Multiscale MHD Flows, II: Minimization of ∇·B Numerical Error

An adaptive numerical dissipation control in a class of high order filter methods for compressible MHD equations is systematically discussed. The filter schemes consist of a divergence-free preserving high order spatial base scheme with a filter approach which can be divergence-free preserving depending on the type of filter operator being used, the method of applying the filter step, and the type of flow problem to be considered. Some of these filter variants provide a natural and efficient way for the minimization of the divergence of the magnetic field (∇·B) numerical error in the sense that commonly used divergence cleaning is not required. Numerical experiments presented emphasize the performance of the ∇·B numerical error. Many levels of grid refinement and detailed comparison of the filter methods with several commonly used compressible MHD shock-capturing schemes will be illustratedA condensed version appears in the Proceedings of the International Conference on High Performance Scientific Computing, March 10-14, 2003, Hanoi, Vietnam. This is a revised version of a longer internal report, Feb. 19, 2004. The longer internal report was published as a RIACS Technical Report TR03.10, July 2003, NASA Ames Research Center     

Non-Linear Filtering and Limiting in High Order Methods for Ideal and Non-Ideal MHD

The adaptive nonlinear filtering and limiting in spatially high order schemes (Yee et al. J. Comput. Phys. 150, 199–238, (1999), Sjögreen and Yee, J. Scient. Comput. 20, 211–255, (2004)) for the compressible Euler and Navier–Stokes equations have been recently extended to the ideal and non-ideal magnetohydrodynamics (MHD) equations, (Sjögreen and Yee, (2003), Proceedings of the 16th AIAA/CFD conference, June 23–26, Orlando F1; Yee and Sjögreen (2003), Proceedings of the International Conference on High Performance Scientific Computing, March, 10–14, Honai, Vietnam; Yee and Sjögreen (2003), RIACS Technical Report TR03. 10, July, NASA Ames Research Center; Yee and Sjögreen (2004), Proceedings of the ICCF03, July 12–16, Toronto, Canada). The numerical dissipation control in these adaptive filter schemes consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free from numerical dissipation contamination. The numerical dissipation considered consists of high order linear dissipation for the suppression of high frequency oscillation and the nonlinear dissipative portion of high-resolution shock-capturing methods for discontinuity capturing. The applicable nonlinear dissipative portion of high-resolution shock-capturing methods is very general. The objective of this paper is to investigate the performance of three commonly used types of discontinuity capturing nonlinear numerical dissipation for both the ideal and non-ideal MHD.     

Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for High Order Methods

The recently developed essentially fourth-order or higher low dissipative shock-capturing scheme of Yee, Sandham, and Djomehri [25] aimed at minimizing numerical dissipations for high speed compressible viscous flows containing shocks, shears and turbulence. To detect non-smooth behavior and control the amount of numerical dissipation to be added, Yee et al. employed an artificial compression method (ACM) of Harten [4] but utilize it in an entirely different context than Harten originally intended. The ACM sensor consists of two tuning parameters and is highly physical problem dependent. To minimize the tuning of parameters and physical problem dependence, new sensors with improved detection properties are proposed. The new sensors are derived from utilizing appropriate non-orthogonal wavelet basis functions and they can be used to completely switch off the extra numerical dissipation outside shock layers. The non-dissipative spatial base scheme of arbitrarily high order of accuracy can be maintained without compromising its stability at all parts of the domain where the solution is smooth. Two types of redundant non-orthogonal wavelet basis functions are considered. One is the B-spline wavelet (Mallat and Zhong [14]) used by Gerritsen and Olsson [3] in an adaptive mesh refinement method, to determine regions where refinement should be done. The other is the modification of the multiresolution method of Harten [5] by converting it to a new, redundant, non-orthogonal wavelet. The wavelet sensor is then obtained by computing the estimated Lipschitz exponent of a chosen physical quantity (or vector) to be sensed on a chosen wavelet basis function. Both wavelet sensors can be viewed as dual purpose adaptive methods leading to dynamic numerical dissipation control and improved grid adaptation indicators. Consequently, they are useful not only for shock-turbulence computations but also for computational aeroacoustics and numerical combustion. In addition, these sensors are scheme independent and can be stand-alone options for numerical algorithms other than the Yee et al. scheme.     

Source Estimation by Full Wave Form Inversion

We consider the inverse problem of estimating the parameters describing the source in a seismic event, using time-dependent ground motion recordings at a number of receiver stations. The inverse problem is defined in terms of a full waveform misfit functional, where the objective function is the integral over time of the weighted $L_2$ distance between observed and synthetic ground motions, summed over all receiver stations. The misfit functional is minimized under the constraint that the synthetic ground motion is governed by the elastic wave equation in a heterogeneous isotropic material. The seismic source is modeled as a point moment tensor forcing in the elastic wave equation. The source is described by 11 parameters: the six unique components of the symmetric moment tensor, the three components of the source location, the origin time, and a frequency parameter modeling the duration of the seismic event. The synthetic ground motions are obtained as the solution of a fourth order accurate finite difference approximation of the elastic wave equation in a heterogeneous isotropic material. The discretization satisfies a summation-by-parts (SBP) property that ensures stability of the explicit time-stepping scheme. We use the SBP property to derive the discrete adjoint of the finite difference method, which is used to efficiently compute the gradient of the misfit. A new moment tensor source discretization is derived that is twice continuously differentiable with respect to the source location. The differentiability makes the Hessian of the misfit a continuous function of all source parameters. We compare four different gradient-based approaches for solving the constrained minimization problem; two non-linear conjugate gradient methods (Fletcher–Reeves and Polak–Ribière), and two quasi-Newton methods (BFGS and L-BFGS). Because the Hessian of the misfit has a very large condition number, the parameters must be scaled before the minimization problem can be solved. Comparing several scaling approaches, we find that the diagonal of the Hessian provides the most reliable scaling alternative. Numerical experiments are presented for estimating the source parameters from synthetic ground motions in two different three-dimensional models; one in a simple layer over half-space, and one using a fully heterogeneous material. Good convergence properties are demonstrated in both cases.     

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High Order Finite Difference and Finite Volume Methods for Advection on the Sphere

Numerical schemes used for computational climate modeling and weather prediction are often of second order accuracy. It is well-known that methods of formal order higher than two offer a significant potential gain in computational efficiency. We here present two classes of high order methods for discretization on the surface of a sphere, first finite difference schemes satisfying the summation-by-parts property on the cube sphere grid, secondly finite volume discretizations on unstructured grids with polygonal cells. Furthermore, we also implement the seventh order accurate weighted essentially non-oscillatory (WENO7) scheme for the cube sphere grid. For the finite difference approximation, we prove a stability estimate, derived from projection boundary conditions. For the finite volume method, we develop the implementational details by working in a local coordinate system at each cell. We apply the schemes to compute advection on a sphere, which is a well established test problem. We compare the performance of the methods with respect to accuracy, computational efficiency, and ability to capture discontinuities.     

A Fourth Order Accurate Finite Difference Scheme for the Elastic Wave Equation in Second Order Formulation

We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient, (??(x)u _x ) _x . This discretization satisfies a summation by parts identity that guarantees stability of the scheme. The boundary conditions are enforced through ghost points, thereby avoiding projections or penalty terms, which often are used with previous summation by parts operators. The temporal discretization is obtained by an explicit modified equation method. Numerical examples with free surface boundary conditions show that the scheme is stable for CFL-numbers up to 1.3, and demonstrate a significant improvement in efficiency over the second order accurate method. The new discretization of (??(x)u _x ) _x has general applicability, and will enable stable fourth order accurate approximations of other partial differential equations as well as the elastic wave equation.     

浅谈钢铁企业封闭料场的建筑设计

本文简要介绍了钢铁企业封闭料场的特点、结构选型及建筑设计方案,为钢铁企业封闭料场的建筑设计提供了参考。     

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