Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations

This paper introduces tensorial calculus techniques in the framework of POD to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree p. Such nonlinear terms have an online complexity of , where k is the dimension of POD basis and therefore is independent of full space dimension. However, it is efficient only for quadratic nonlinear terms because for higher nonlinearities, POD model proves to be less time consuming once the POD basis dimension k is increased. Numerical experiments are carried out with a two‐dimensional SWE test problem to compare the performance of tensorial POD, POD, and POD/discrete empirical interpolation method (DEIM). Numerical results show that tensorial POD decreases by 76× the computational cost of the online stage of POD model for configurations using more than 300,000 model variables. The tensorial POD SWE model was only 2 to 8× slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM SWE model to compute its offline stage faster than POD and tensorial POD approaches. Copyright © 2014 John Wiley & Sons, Ltd.     

2D Burgers equation with large Reynolds number using POD/DEIM and calibration

Model order reduction of the two‐dimensional Burgers equation is investigated. The mathematical formulation of POD/discrete empirical interpolation method (DEIM)‐reduced order model (ROM) is derived based on the Galerkin projection and DEIM from the existing high fidelity‐implicit finite‐difference full model. For validation, we numerically compared the POD ROM, POD/DEIM, and the full model in two cases of Re = 100 and Re = 1000, respectively. We found that the POD/DEIM ROM leads to a speed‐up of CPU time by a factor of O(10). The computational stability of POD/DEIM ROM is maintained by means of a careful selection of POD modes and the DEIM interpolation points. The solution of POD/DEIM in the case of Re = 1000 has an accuracy with error O(10^?3) versus O(10^?4) in the case of Re = 100 when compared with the high fidelity model. For this turbulent flow, a closure model consisting of a Tikhonov regularization is carried out in order to recover the missing information and is developed to account for the small‐scale dissipation effect of the truncated POD modes. It is shown that the computational results of this calibrated ROM exhibit considerable agreement with the high fidelity model, which implies the efficiency of the closure model used. Copyright © 2016 John Wiley & Sons, Ltd.     

Reduced‐order modeling based on POD of a parabolized Navier–Stokes equation model I: forward model

A proper orthogonal decomposition (POD)‐based reduced‐order model of the parabolized Navier–Stokes (PNS) equations is derived in this article. A space‐marching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to construct the reduced‐order model. In order to improve the accuracy and the stability of the reduced‐order model in the presence of a high Reynolds number, we applied a Sobolev H₁ norm calibration to the POD construction process. Finally, some numerical tests with a high‐fidelity model as well as the POD reduced‐order model were carried out to demonstrate the efficiency and the accuracy of the reduced‐order model for solving the PNS equations compared with the full PNS model. Different inflow conditions and different selections of snapshots were experimented to test the POD reduction technique. The efficiency of the H₁ norm POD calibration is illustrated for the PNS model with increasingly higher Reynolds numbers, along with the optimal dissipation coefficient derivation, yielding the best root mean square error and correlation coefficient between the full and reduced‐order PNS models. Copyright © 2011 John Wiley & Sons, Ltd.     

An efficient numerical method for solving Falkner–Skan boundary layer flows

In this paper a novel computational technique for the solution of nonlinear third‐order boundary value problems is presented. We demonstrate the application of the method by solving the famous Falkner–Skan equation on a semi‐infinite domain. Comparison with the results from other methods such as the homotopy analysis method and numerical methods demonstrates the accuracy, computational efficiency and robustness of this technique. Copyright © 2011 John Wiley & Sons, Ltd.     

Non‐intrusive reduced order modelling with least squares fitting on a sparse grid

This paper presents a non‐intrusive reduced order model for general, dynamic partial differential equations. Based upon proper orthogonal decomposition (POD) and Smolyak sparse grid collocation, the method first projects the unknowns with full space and time coordinates onto a reduced POD basis. Then we introduce a new least squares fitting procedure to approximate the dynamical transition of the POD coefficients between subsequent time steps, taking only a set of full model solution snapshots as the training data during the construction. Thus, neither the physical details nor further numerical simulations of the original PDE model are required by this methodology, and the level of non‐intrusiveness is improved compared with existing reduced order models. Furthermore, we take adaptive measures to address the instability issue arising from reduced order iterations of the POD coefficients. This model can be applied to a wide range of physical and engineering scenarios, and we test it on a couple of problems in fluid dynamics. It is demonstrated that this reduced order approach captures the dominant features of the high‐fidelity models with reasonable accuracy while the computation complexity is reduced by several orders of magnitude. Copyright © 2016 John Wiley & Sons, Ltd.     

A new multistage spectral relaxation method for solving chaotic initial value systems

In this paper, we present a new pseudospectral method application for solving nonlinear initial value problems (IVPs) with chaotic properties. The proposed method, called the multistage spectral relaxation method (MSRM) is based on a novel technique of extending Gauss–Siedel type relaxation ideas to systems of nonlinear differential equations and using the Chebyshev pseudo-spectral methods to solve the resulting system on a sequence of multiple intervals. In this new application, the MSRM is used to solve famous chaotic systems such as the such as Lorenz, Chen, Liu, Rikitake, Rössler, Genesio–Tesi, and Arneodo–Coullet chaotic systems. The accuracy and validity of the proposed method is tested against Runge–Kutta and Adams–Bashforth–Moulton based methods. The numerical results indicate that the MSRM is an accurate, efficient, and reliable method for solving very complex IVPs with chaotic behavior.     

Model reduction for reacting flow applications

A model reduction approach based on Galerkin projection, proper orthogonal decomposition (POD), and the discrete empirical interpolation method (DEIM) is developed for chemically reacting flow applications. Such applications are challenging for model reduction due to the strong coupling between fluid dynamics and chemical kinetics, a wide range of temporal and spatial scales, highly nonlinear chemical kinetics, and long simulation run-times. In our approach, the POD technique combined with Galerkin projection reduces the dimension of the state (unknown chemical concentrations over the spatial domain), while the DEIM approximates the nonlinear chemical source term. The combined method provides an efficient offline–online solution strategy that enables rapid solution of the reduced-order models. Application of the approach to an ignition model of a premixed H₂/O₂/Ar mixture with 19 reversible chemical reactions and 9 species leads to reduced-order models with state dimension several orders of magnitude smaller than the original system. For example, a reduced-order model with state dimension of 60 accurately approximates a full model with a dimension of 91,809. This accelerates the simulation of the chemical kinetics by more than two orders of magnitude. When combined with the full-order flow solver, this results in a reduction of the overall computational time by a factor of approximately 10. The reduced-order models are used to analyse the sensitivity of outputs of interest with respect to uncertain input parameters describing the reaction kinetics.     

基于特征正交分解的一类瞬态非线性热传导问题的新型快速分析方法

提出了一种基于特征正交分解(POD)和有限元法的瞬态非线性热传导问题的模型降阶快速分析方法, 建立了导热系数随温度变化的一类瞬态非线性热传导问题有限元格式的POD降阶模型. 在隐式时间推进方法的基础上有效结合单元预转换方法和多级线性化方法发展了一种加速求解瞬态非线性热传导降阶模型的新型计算方法,并通过二维和三维算例验证了该方法的准确性和高效性. 研究结果表明: (1)降阶模型解的均方根误差在经过初始时段轻微的脉动后稳定于0.01%以下, 而其计算效率比有限元全阶模型提高2$\sim $3个数量级, 并且自由度数量(DOFs)愈大提高的幅度也愈加显著; (2)新型算法解决了常规算法在计算非线性降阶模型时加速性能差的问题, 即使是在DOFs比较小的时候也能够明显提高计算效率; (3)常数边界条件下得到的POD模态可以用来建立相同求解域在各种复杂时变边界条件下的瞬态非线性热传导降阶模型, 并对其传热过程和温度场进行快速准确的分析与预测, 具有很好的工程应用价值.      

Principal interval decomposition framework for POD reduced‐order modeling of convective Boussinesq flows

A principal interval decomposition (PID) approach is presented for the reduced‐order modeling of unsteady Boussinesq equations. The PID method optimizes the lengths of the time windows over which proper orthogonal decomposition (POD) is performed and can be highly effective in building reduced‐order models for convective problems. The performance of these POD models with and without using the PID approach is investigated by applying these methods to the unsteady lock‐exchange flow problem. This benchmark problem exhibits a strong shear flow induced by a temperature jump and results in the Kelvin–Helmholtz instability. This problem is considered a challenging benchmark problem for the development of reduced‐order models. The reference solutions are obtained by direct numerical simulations of the vorticity and temperature transport equations using a compact fourth‐order‐accurate scheme. We compare the accuracy of reduced‐order models developed with different numbers of POD basis functions and different numbers of principal intervals. A linear interpolation model is constructed to obtain basis functions when varying physical parameters. The predictive performance of our models is then analyzed over a wide range of Reynolds numbers. It is shown that the PID approach provides a significant improvement in accuracy over the standard Galerkin POD reduced‐order model. This numerical assessment of the PID shows that it may represent a reliable model reduction tool for convection‐dominated, unsteady‐flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

On the solution of MHD flow over a nonlinear stretching sheet by an efficient semi‐analytical technique

The problem of magneto‐hydrodynamic fluid flow past a nonlinear stretching sheet in the presence of a transverse magnetic field is analyzed. The governing equations are transformed into a nonlinear ordinary differential equation that is solved using a novel spectral homotopy analysis method and the Matlab in‐built numerical solverttbvp4c. The new technique removes some known limitations of the homotopy analysis method and offers a more systematic way of selecting initial approximations and the optimal auxiliary parameter ?. A comparison with the numerical solution confirms the robustness, the computational efficiency, and the accuracy of the technique. Copyright © 2011 John Wiley & Sons, Ltd.     

A monotone finite volume scheme for advection–diffusion equations on distorted meshes

A new monotone finite volume method with second‐order accuracy is presented for the steady‐state advection–diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes that guarantee the positivity of the numerical solution. The approximation of the diffusive flux is based on nonlinear two‐point approximation, and the approximation of the advective flux is based on the second‐order upwind method with proper slope limiter. The second‐order convergence rate for concentration and the monotonicity of the nonlinear finite volume method are verified with numerical experiments. Copyright © 2011 John Wiley & Sons, Ltd.     

Condensed Galerkin element of degree m for first-order initial-value problem with O(h2m+2) super-convergent nodal solutions

A new type of Galerkin finite element for first-order initial-value problems (IVPs) is proposed. Both the trial and test functions employ the same m-degreed polynomials. The adjoint equation is used to eliminate one degree of freedom (DOF) from the test function, and then the so-called condensed test function and its consequent condensed Galerkin element are constructed. It is mathematically proved and numerically verified that the condensed element produces the super-convergent nodal solutions of O(h^2m+2), which is equivalent to the order of accuracy by the conventional element of degree m + 1. Some related properties are addressed, and typical numerical examples of both linear and nonlinear IVPs of both a single equation and a system of equations are presented to show the validity and effectiveness of the proposed element.

     

非线性振动系统周期运动及其稳定性的数值研究

§1引言确定型非线性振动系统的运动可分类如下: 1.非定常运动;2.定常运动:(1)周期运动,(2)各态历经运动,(3)浑沌运动。其中非定常运动是一暂态过程,会随着时间的增长逐步衰减乃至实际上消失。定常运动中的各态历经运动,指系统至少有两个互不通约(即其比值为无理数)的振动频率,因此运动虽然局      

Multi‐material incompressible flow simulation using the moment‐of‐fluid method

This paper compares the numerical performance of the moment‐of‐fluid (MOF) interface reconstruction technique with Youngs, LVIRA, power diagram (PD), and Swartz interface reconstruction techniques in the context of a volume‐of‐fluid (VOF) based finite element projection method for the numerical simulation of variable‐density incompressible viscous flows. In pure advection tests with multiple materials MOF shows dramatic improvements in accuracy compared with the other methods. In incompressible flows where density differences determine the flow evolution, all the methods perform similarly for two material flows on structured grids. On unstructured grids, the second‐order MOF, LVIRA, and Swartz methods perform similarly and show improvement over the first‐order Youngs' and PD methods. For flow simulations with more than two materials, MOF shows increased accuracy in interface positions on coarse meshes. In most cases, the convergence and accuracy of the computed flow solution was not strongly affected by interface reconstruction method. Published in 2009 by John Wiley & Sons, Ltd.     

An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model

Proper orthogonal decomposition (POD) and singular value decomposition (SVD) methods are used to study a finite difference discretization scheme (FDS) for the tropical Pacific Ocean reduced gravity model. Ensembles of data are compiled from transient solutions computed from the discrete equation system derived by FDS for the tropical Pacific Ocean reduced gravity model. The optimal orthogonal bases are used to reconstruct the elements of the ensemble with POD and SVD. Combining the above approach with a Galerkin projection procedure yields a new optimizing FDS model of lower dimensions and high accuracy for the tropical Pacific Ocean reduced gravity model. An error estimate of the new reduced order optimizing FDS model is then derived. Numerical examples are presented illustrating that the error between the POD approximate solution and the full FDS solution is consistent with previously obtained theoretical results, thus validating the feasibility and efficiency of POD method. Copyright © 2007 John Wiley & Sons, Ltd.     

Verification of a mixed high‐order accurate DNS code for laminar turbulent transition by the method of manufactured solutions

This paper presents results on a verification test of a Direct Numerical Simulation code of mixed high‐order of accuracy using the method of manufactured solutions (MMS). This test is based on the formulation of an analytical solution for the Navier–Stokes equations modified by the addition of a source term. The present numerical code was aimed at simulating the temporal evolution of instability waves in a plane Poiseuille flow. The governing equations were solved in a vorticity–velocity formulation for a two‐dimensional incompressible flow. The code employed two different numerical schemes. One used mixed high‐order compact and non‐compact finite‐differences from fourth‐order to sixth‐order of accuracy. The other scheme used spectral methods instead of finite‐difference methods for the streamwise direction, which was periodic. In the present test, particular attention was paid to the boundary conditions of the physical problem of interest. Indeed, the verification procedure using MMS can be more demanding than the often used comparison with Linear Stability Theory. That is particularly because in the latter test no attention is paid to the nonlinear terms. For the present verification test, it was possible to manufacture an analytical solution that reproduced some aspects of an instability wave in a nonlinear stage. Although the results of the verification by MMS for this mixed‐order numerical scheme had to be interpreted with care, the test was very useful as it gave confidence that the code was free of programming errors. Copyright © 2009 John Wiley & Sons, Ltd.     

非线性转子-轴承系统的周期解及近似解析表达式

通过对普通打靶方法进行改造提出一种确定非线性系统周期轨道及周期的新型打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参与打靶法的迭代过程，迭代过程包含对周期轨道和周期的求解，迭代过程中的增量通过优化方法选择，从而能迅速确定出系统的周期轨道及其周期。应用所求的结果结合谐波平衡方法求得了非线性系统的周期轨道的近似解析表达式，理论上通过增加谐波的阶数任何精度的周期解都可以得到。最后将该方法应用于非线性转子轴承系统，求出了在某些参数下转子的周期解及其近似解析表达式，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性，计算结果对于转子系统运动的定量控制有重要理论指导意义。     

Analysis of efficient preconditioned defect correction methods for nonlinear water waves

Robust computational procedures for the solution of non‐hydrostatic, free surface, irrotational and inviscid free‐surface water waves in three space dimensions can be based on iterative preconditioned defect correction (PDC) methods. Such methods can be made efficient and scalable to enable prediction of free‐surface wave transformation and accurate wave kinematics in both deep and shallow waters in large marine areas or for predicting the outcome of experiments in large numerical wave tanks. We revisit the classical governing equations are fully nonlinear and dispersive potential flow equations. We present new detailed fundamental analysis using finite‐amplitude wave solutions for iterative solvers. We demonstrate that the PDC method in combination with a high‐order discretization method enables efficient and scalable solution of the linear system of equations arising in potential flow models. Our study is particularly relevant for fast and efficient simulation of non‐breaking fully nonlinear water waves over varying bottom topography that may be limited by computational resources or requirements. To gain insight into algorithmic properties and proper choices of discretization parameters for different PDC strategies, we study systematically limits of accuracy, convergence rate, algorithmic and numerical efficiency and scalability of the most efficient known PDC methods. These strategies are of interest, because they enable generalization of geometric multigrid methods to high‐order accurate discretizations and enable significant improvement in numerical efficiency while incuring minimal storage requirements. We demonstrate robustness using such PDC methods for practical ranges of interest for coastal and maritime engineering, that is, from shallow to deep water, and report details of numerical experiments that can be used for benchmarking purposes. Copyright © 2014 John Wiley & Sons, Ltd.     

An adaptive cut‐cell method for environmental fluid mechanics

In this work we present a numerical method for solving the incompressible Navier–Stokes equations in an environmental fluid mechanics context. The method is designed for the study of environmental flows that are multiscale, incompressible, variable‐density, and within arbitrarily complex and possibly anisotropic domains. The method is new because in this context we couple the embedded‐boundary (or cut‐cell) method for complex geometry with block‐structured adaptive mesh refinement (AMR) while maintaining conservation and second‐order accuracy. The accurate simulation of variable‐density fluids necessitates special care in formulating projection methods. This variable‐density formulation is well known for incompressible flows in unit‐aspect ratio domains, without AMR, and without complex geometry, but here we carefully present a new method that addresses the intersection of these issues. The methodology is based on a second‐order‐accurate projection method with high‐order‐accurate Godunov finite‐differencing, including slope limiting and a stable differencing of the nonlinear convection terms. The finite‐volume AMR discretizations are based on two‐way flux matching at refinement boundaries to obtain a conservative method that is second‐order accurate in solution error. The control volumes are formed by the intersection of the irregular embedded boundary with Cartesian grid cells. Unlike typical discretization methods, these control volumes naturally fit within parallelizable, disjoint‐block data structures, and permit dynamic AMR coarsening and refinement as the simulation progresses. We present two‐ and three‐dimensional numerical examples to illustrate the accuracy of the method. Copyright © 2008 John Wiley & Sons, Ltd.