Sound speed profiles in high spatiotemporal resolution using multigrid three-dimensional variational method:a coastal experiment off northern Shandong Peninsula

It is essential to ac quire sound speed profiles(SSPs)in high-precision spatiotemporal resolution for undersea acoustic activities.However,conventional observation methods cannot obtain high-resolution SSPs.Besides,S SPs are complex and changeable in time and space,especially in coastal areas.We proposed a new space-time multigrid three-dimensional variational method with weak constraint term(referred to as STC-MG3DVar)to construct high-precision spatiotemporal resolution SSPs in coastal areas,in which sound velocity is defined as the analytical variable,and the Chen-Millero sound velocity empirical formula is introduced as a weak constraint term into the cost function of the STC-MG3DVar.The spatiotemporal correlation of sound velocity observations is taken into account in the STC-MG3DVar method,and the multi-scale information of sound velocity observations from long waves to short waves can be successively extracted.The weak constraint term can optimize sound velocity by the physical relationship between sound velocity and temperature-salinity to obtain more reasonable and accurate SSPs.To verify the accuracy of the STC-MG3DVar,SSPs observations and CTD observations(temperature observations,salinity observations)are obtained from field experiments in the northern coastal area of the Shandong Peninsula.The average root mean square error(RMSE)of the STC-MG3DVar-constructed SSPs is 0.132 m/s,and the STC-MG3DVar method can improve the SSPs construction accuracy over the space-time multigrid 3DVar without weak constraint term(ST-MG3DVar)by 10.14%and over the spatial multigrid 3DVar with weak constraint term(SC-MG3DVar)by 44.19%.With the advantage of the constraint term and the spatiotemporal correlation information,the proposed STC-MG3DVar method works better than the ST-MG3DVar and the SCMG3DVar in constructing high-precision spatiotemporal re solution SSPs.

A New Extrapolation Economy Cascadic Multigrid Method for Image Restoration Problems

In this paper, a new extrapolation economy cascadic multigrid method is proposed to solve the image restoration model. The new method combines the new extrapolation formula and quadratic interpolation to design a nonlinear prolongation operator, which provides more accurate initial values for the fine grid level. An edge preserving denoising operator is constructed to remove noise and preserve image edges. The local smoothing operator reduces the influence of staircase effect. The experiment results show that the new method not only improves the computational efficiency but also ensures good recovery quality.

基于5阶精度格式WCNS-E-5的p-multigrid方法研究

p-multigrid方法的基本思想是:在保证收敛结果为高阶精度的同时,利用低阶精度格式耗散大的特点,来改善高精度有限差分格式在迭代计算时收敛速度慢的弱点。本文基于5阶精度WCNS-E-5差分格式,引入1阶精度迎风格式和3阶精度加权格式,构造了p-multigrid方法,在迭代过程中采用了V循环、W循环、S循环、Pre_V和FMG循环等不同方式来应用这三种格式,并通过典型算例考察了这些循环方式对收敛速度的影响,初步数值试验表明,采用恰当的循环方式,本文所设计的p-multigrid方法能够加快收敛速度,并保证了最终收敛结果与5阶精度WCNS-E-5差分格式的一致性。

A Preconditioned Multigrid Method for Efficient Simulation of Three-dimensional Compressible and Incompressible Flows

To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Rtmge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.

A theoretical study of the multigrid three-dimensional variational data assimilation scheme using a simple bilinear interpolation algorithm

In order to solve the so-called ＂bull-eye＂ problem caused by using a simple bilinear interpolation as an observational mapping operator in the cost function in the multigrid three-dimensional variational （3DVAR） data assimilation scheme, a smoothing term, equivalent to a penalty term, is introduced into the cost function to serve as a means of troubleshooting. A theoretical analysis is first performed to figure out what on earth results in the issue of ＂bull-eye＂, and then the meaning of such smoothing term is elucidated and the uniqueness of solution of the multigrid 3DVAR with the smoothing term added is discussed through the theoretical deduction for one-dimensional （1D） case, and two idealized data assimilation experiments （one- and two-dimensional （2D） cases）. By exploring the relationship between the smoothing term and the recursive filter theoretically and practically, it is revealed why satisfied analysis results can be achieved by using such proposed solution for the issue of the multigrid 3DVAR.

MULTIGRID TECHNIQUE APPLIED TO LINEAR TIME-DEPENDENT HYPERBOLIC SYSTEMS

A system of linear time-dependent hyperbolic partial differential equations in the form of the time-domain Maxwell′s equations is numerically solved using ageometric multigrid method.The multilevel method is an adaptation of Ni′s cell-vertex based multigrid technique,originally proposed for accelerating steady state convergence of nonlinear time-dependent Euler equations of gas dynamics.We discuss issues pertaining to the application of the geometric multigrid method to a system of equations where the major issue is of accurately propagating linear waves over large distances leading to major constraints on the required grid resolution in terms of points-perwavelength.

L^2-ERROR OF EXTRAPOLATION CASCADIC MULTIGRID (EXCMG)

Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method （EXCMG） is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of multiple grids, both superconvergence error in H^1-norm and the optimal error in l2-norm are analyzed. The numerical experiment shows the advantage of EXCMG in comparison with CMG.

System of Multigrid Nonlinear Least-squares Four-dimensional Variational Data Assimilation for Numerical Weather Prediction(SNAP):System Formulation and Preliminary Evaluation

A new forecasting system-the System of Multigrid Nonlinear Least-squares Four-dimensional Variational(NLS-4DVar)Data Assimilation for Numerical Weather Prediction(SNAP)-was established by building upon the multigrid NLS-4DVar data assimilation scheme,the operational Gridpoint Statistical Interpolation(GSI)−based data-processing and observation operators,and the widely used Weather Research and Forecasting numerical model.Drawing upon lessons learned from the superiority of the operational GSI analysis system,for its various observation operators and the ability to assimilate multiple-source observations,SNAP adopts GSI-based data-processing and observation operator modules to compute the observation innovations.The multigrid NLS-4DVar assimilation framework is used for the analysis,which can adequately correct errors from large to small scales and accelerate iteration solutions.The analysis variables are model state variables,rather than the control variables adopted in the conventional 4DVar system.Currently,we have achieved the assimilation of conventional observations,and we will continue to improve the assimilation of radar and satellite observations in the future.SNAP was evaluated by case evaluation experiments and one-week cycling assimilation experiments.In the case evaluation experiments,two six-hour time windows were established for assimilation experiments and precipitation forecasts were verified against hourly precipitation observations from more than 2400 national observation sites.This showed that SNAP can absorb observations and improve the initial field,thereby improving the precipitation forecast.In the one-week cycling assimilation experiments,six-hourly assimilation cycles were run in one week.SNAP produced slightly lower forecast RMSEs than the GSI 4DEnVar(Four-dimensional Ensemble Variational)as a whole and the threat scores of precipitation forecasts initialized from the analysis of SNAP were higher than those obtained from the analysis of GSI 4DEnVar.

Biorthogonal Wavelet Based Algebraic Multigrid Preconditioners for Large Sparse Linear Systems

In this article algebraic multigrid as preconditioners are designed, with biorthogonal wavelets, as intergrid operators for the Krylov subspace iterative methods. Construction of hierarchy of matrices in algebraic multigrid context is based on lowpass filter version of Wavelet Transform. The robustness and efficiency of this new approach is tested by applying it to large sparse, unsymmetric and ill-conditioned matrices from Tim Davis collection of sparse matrices. Proposed preconditioners have potential in reducing cputime, operator complexity and storage space of algebraic multigrid V-cycle and meet the desired accuracy of solution compared with that of orthogonal wavelets.

An efficient multigrid-DEIM semi-reduced-order model for simulation of single-phase compressible flow in porous media

In this paper,an efcient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible fow in porous media.The cornerstone of the proposed model is that the full approximate storage multigrid method is used to accelerate the solution of fow equation in original full-order space,and the discrete empirical interpolation method(DEIM)is applied to speed up the solution of Peng-Robinson equation of state in reduced-order subspace.The multigrid-DEIM semi-reduced-order model combines the computation both in full-order space and in reducedorder subspace,which not only preserves good prediction accuracy of full-order model,but also gains dramatic computational acceleration by multigrid and DEIM.Numerical performances including accuracy and acceleration of the proposed model are carefully evaluated by comparing with that of the standard semi-implicit method.In addition,the selection of interpolation points for constructing the low-dimensional subspace for solving the Peng-Robinson equation of state is demonstrated and carried out in detail.Comparison results indicate that the multigrid-DEIM semi-reduced-order model can speed up the simulation substantially at the same time preserve good computational accuracy with negligible errors.The general acceleration is up to 50-60 times faster than that of standard semi-implicit method in two-dimensional simulations,but the average relative errors of numerical results between these two methods only have the order of magnitude 10^(−4)-10^(−6)%.

A New Multigrid 3D-VAR Optimization Method for Bottom Friction Using HF Radar Current Observation

This paper proposes a new method for data assimilation of the surface radial current observed by High Frequency ground wave radar and optimization of the bottom friction coefficient.In this method,the shallow water wave equation is introduced into the cost function of the multigrid three-dimensional variation data assimilation method as the weak constraint term,the surface current and the bottom friction coefficient are defined as the analytical variables,and the high spatiotemporal resolution surface radial flow observed by the high-frequency ground wave radar is used to optimize the surface current and bottom friction coefficient.This method can effectively consider the spatiotemporal correlation of radar data and extract multiscale information from surface radial flow data from long waves to short waves.Introducing the shallow water wave equation into the cost function as a weak constraint condition can adjust both the momentum and mass fields simultaneously to obtain more reasonable analysis information.The optimized bottom friction coefficient is introduced into the regional ocean numerical model to carry out numerical experiments.The test results show that the bottom friction coefficient obtained by this method can effectively improve the accuracy of the numerical simulation of sea surface height in the offshore area and reduce the simulation error.

Multigrid ILU Panel Method for the Calculation of Wavemaking Resistance

Panel methods for the calculation of wavemaking resistance result in a linear equation system for the unknown singularities.The coefficient matrix is full but not well conditioned.In this paper an incomplete LU decomposition (ILU) method and a combined multigrid ILU method are used to solve the linear system.Systematic computations using the ILU method have shown that the CPU time can be reduced to 30% to 40% of that using an incomplete Gaussian elimination method. In the proposed multigrid ILU method an averaged restriction and a piecewise constant prolongation are used.The construction of the coefficient matrix at coarse levels is based on geometrical considerations.It turns out that the condition of the relative consistency is fulfilled.Comparison computations have shown that nearly the same results were obtained.However,due to additional CPU time needed for the execution of the matrix vector products in the restriction and the prolongation proceses of the multigrid method,a further reduction of the total CPU time could not be reailized.

AN OPTIMAL V-CYCLE MULTIGRID METHOD FOR CONFORMING AND NONCONFORMING PLATE ELEMENTS

In this paper, an optimal V-cycle multigrid method for some conforming and nonconforming plate elements are constructed. A new method dealing with nonnested multigrid methods is presented.

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

A Newton multigrid method is developed for one-dimensional （1D） and two- dimensional （2D） steady-state shallow water equations （SWEs） with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady- state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.

Multigrid Solution of an Elliptic Fredholm Partial Integro-Differential Equation with a Hilbert-Schmidt Integral Operator

An efficient multigrid finite-differences scheme for solving elliptic Fredholm partial integro-differential equations (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization of the PIDE problem with a multigrid scheme that includes a fast multilevel integration of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework.

The V-cycle Multigrid Method for a Hermite-type Rectangular Element

在这份报纸，一个 V 周期 multigrid 方法为一个 Hermite 矩形的元素被介绍。由定义合适的网孔依赖者内部产品和转移操作员，我们获得它的集中性质和网孔尺寸和水平的一致集中率独立人士被建立。

The Numerical Solution of the MRLW Equation Using the Multigrid Method

In this paper, we obtained the numerical solutions of the modified regularized long-wave (MRLW) equation, by using the multigrid method and finite difference method. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Usingerror norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other techniques.

Multigrid One-Shot Method for PDE-Constrained Optimization Problems

This paper presents a numerical method for PDE-constrained optimization problems. These problems arise in many fields of science and engineering including those dealing with real applications. The physical problem is modeled by partial differential equations (PDEs) and involve optimization of some quantity. The PDEs are in most cases nonlinear and solved using numerical methods. Since such numerical solutions are being used routinely, the recent trend has been to develop numerical methods and algorithms so that the optimization problems can be solved numerically as well using the same PDE-solver. We present here one such numerical method which is based on simultaneous pseudo-time stepping. The efficiency of the method is increased with the help of a multigrid strategy. Application example is included for an aerodynamic shape optimization problem.

Geometric Multigrid Method for Isogeometric Analysis

The isogeometric analysis method(IGA)is a new type of numerical method solving partial differential equations.Compared with the traditional finite element method,IGA based on geometric spline can keep the model consistency between geometry and analysis,and provide higher precision with less freedom.However,huge stiffness matrix fromthe subdivision progress still leads to the solution efficiency problems.This paper presents amultigrid method based on geometric multigrid(GMG)to solve the matrix system of IGA.This method extracts the required computational data for multigrid method fromthe IGA process,which also can be used to improve the traditional algebraic multigrid method(AGM).Based on this,a full multigrid method(FMG)based on GMG is proposed.In order to verify the validity and reliability of these methods,this paper did some test on Poisson’s equation and Reynolds’equation and compared the methods on different subdivision methods,different grid degrees of freedom,different cyclic structure degrees,and studied the convergence rate under different subdivision strategies.The results show that the proposed method is superior to the conventional algebraic multigrid method,and for the standard relaxed V-cycle iteration,the method still has a convergence speed independent of the grid size at the same degrees.

THE MULTIGRID METHOD FOR RESERVOIR SIMULATION

This paper describes a way of solving the reservoir simulation pressure equation using mulligrid technique. The subroutine MG of four-grid method is presented. The result for 2-D two-phase problem is exactly the same as that of the SOR method and the CPU time is much less than that of the latter one.