Discretization Of Partial Differential Equations Research Articles

On the use of modal derivatives for nonlinear model order reduction

SummaryModal derivative is an approach to compute a reduced basis for model order reduction of large‐scale nonlinear systems that typically stem from the discretization of partial differential equations. In this way, a complex nonlinear simulation model can be integrated into an optimization problem or the design of a controller, based on the resulting small‐scale state‐space model. We investigate the approximation properties of modal derivatives analytically and thus lay a theoretical foundation of their use in model order reduction, which has been missing so far. Concentrating on the application field of structural mechanics and structural dynamics, we show that the concept of modal derivatives can also be applied as nonlinear extension of the Craig–Bampton family of methods for substructuring. We furthermore generalize the approach from a pure projection scheme to a novel reduced‐order modeling method that replaces all nonlinear terms by quadratic expressions in the reduced state variables. This complexity reduction leads to a frequency‐preserving nonlinear quadratic state‐space model. Numerical examples with carefully chosen nonlinear model problems and three‐dimensional nonlinear elasticity confirm the analytical properties of the modal derivative reduction and show the potential of the proposed novel complexity reduction methods, along with the current limitations. Copyright © 2016 John Wiley & Sons, Ltd.

Approximation error estimates and inverse inequalities for B-splines of maximum smoothness

In this paper, we develop approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard Sobolev norms and semi-norms. The presented approximation error estimates do not depend on the polynomial degree of the splines but only on the grid size. We will see that the approximation lives in a subspace of the classical B-spline space. We show that for this subspace, there is an inverse inequality which is also independent of the polynomial degree. As the approximation error estimate and the inverse inequality show complementary behavior, the results shown in this paper can be used to construct fast iterative methods for solving problems arising from isogeometric discretizations of partial differential equations.

The Joy and Pain of Skew Symmetry

In this paper, we review recent progress on two related issues. Firstly, the discretisation of partial differential equations of quantum mechanics in a semiclassical regime. Due to the presence of a small parameter, such equations exhibit high oscillations and multiscale behaviour, rendering them difficult to discretise. We describe a methodology, using symmetric Zassenhaus splittings in a free Lie algebra, which allows for their exceedingly fast and accurate numerics. The imperative of preserving the unitarity of the underlying flow takes us to the second theme of this paper, approximation of derivatives by skew-symmetric matrices. Here, we identify a gap in the elementary theory of finite-difference approximations: in the presence of Dirichlet boundary conditions, it is impossible to approximate the derivative to order higher than two on a uniform grid! This motivates the investigation of skew symmetry on non-uniform grids, an endeavour which, although still in its infancy, is already replete with interesting results. We conclude by discussing a number of generalisations and open problems.

On numerical integration in isogeometric subdivision methods for PDEs on surfaces

Subdivision surfaces offer great flexibility in capturing irregular topologies combined with higher order smoothness. For instance, Loop and Catmull–Clark subdivision schemes provide C2 smoothness everywhere except at extraordinary vertices, where the generated surfaces are C1 smooth. The combination of flexibility and smoothness leads to their frequent use in geometric modeling and makes them an ideal candidate for performing isogeometric analysis of higher order problems with irregular topologies, e.g. thin shell problems. In the neighborhood of extraordinary vertices, however, the resulting surfaces (and the functions defined on them) are non-polynomial. Thus numerical integration based on quadrature has to be performed carefully to preserve the expected higher order consistency.This paper presents a detailed case study of different quadrature schemes for isogeometric discretizations of partial differential equations on closed surfaces with Loop’s subdivision scheme. As model problems we consider elliptic equations with the Laplace–Beltrami and the surface bi-Laplacian operator as well as the associated eigenvalue problem for the Laplace–Beltrami operator. A particular emphasis is on the robustness of the approach in the vicinity of extraordinary vertices. Based on a series of numerical experiments, different quadrature schemes are compared. Mid-edge quadrature, which can easily be implemented via a lookup table, turns out to be a preferable choice due to its robustness and efficiency.

A minimum Sobolev norm technique for the numerical discretization of PDEs

Partial differential equations (PDEs) are discretized into an under-determined system of equations and a minimum Sobolev norm solution is shown to be efficient to compute and converge under very generic conditions. Numerical results of a single code, that can handle PDEs in first-order form on complicated polygonal geometries, are shown for a variety of PDEs: variable coefficient div–curl, scalar elliptic PDEs, elasticity equation, stationary linearized Navier–Stokes, scalar fourth-order elliptic PDEs, telegrapher's equations, singular PDEs, etc.

An efficient numerical partial differential equation approach for pricing foreign exchange interest rate hybrid derivatives

In this paper, we discuss efficient pricing methods via a partial differential equation (PDE) approach for long-dated foreign exchange (FX) interest rate hybrids under a three-factor multicurrency pricing model with FX volatility skew. The emphasis of this paper is on power-reverse dual-currency (PRDC) swaps with popular exotic features, namely knockout and FX target redemption (FX-TARN). Challenges in pricing these derivatives via a PDE approach arise from the high dimensionality of the model PDE as well as the complexities in handling the exotic features, especially in the case of the FX-TARN provision, due to its path dependency. Our proposed PDE pricing framework for FX-TARN PRDC swaps is based on partitioning the pricing problem into several independent pricing subproblems over each time period of the swap's tenor structure, with possible communication at the end of the time period. Each of these pricing subproblems can be viewed as equivalent to a knockout PRDC swap with a known time-dependent barrier and requires a solution of the model PDE, which, in our case, is a time-dependent parabolic PDE in three space dimensions. Finite difference schemes on nonuniform grids are used for the spatial discretization of the model PDE, and the alternating direction implicit time-stepping methods are employed for its time discretization. Numerical examples illustrating the convergence properties and efficiency of the numerical methods are also provided.

A parallel fast boundary element method using cyclic graph decompositions

We propose a method of a parallel distribution of densely populated matrices arising in boundary element discretizations of partial differential equations. In our method the underlying boundary element mesh consisting of n elements is decomposed into N submeshes. The related N×N submatrices are assigned to N concurrent processes to be assembled. Additionally we require each process to hold exactly one diagonal submatrix, since its assembling is typically most time consuming when applying fast boundary elements. We obtain a class of such optimal parallel distributions of the submeshes and corresponding submatrices by cyclic decompositions of undirected complete graphs. It results in a method the theoretical complexity of which is O((n/N)log(n/N))$O((n/\sqrt {N})\log (n/\sqrt {N}))$ in terms of time for the setup, assembling, matrix action, as well as memory consumption per process. Nevertheless, numerical experiments up to n=2744832 and N=273 on a real-world geometry document that the method exhibits superior parallel scalability O((n/N)logn)$O((n/N)\,\log n)$ of the overall time, while the memory consumption scales accordingly to the theoretical estimate.

Constrained pseudo-transient continuation

This paper addresses the problem of finding a stationary point of a nonlinear dynamical system whose state variables are under inequality constraints. Systems of this type often arise from the discretization of partial differential equations that model physical phenomena (e.g. fluid dynamics) in which the state variables are under realizability constraints (e.g. positive pressure and density). We start from the popular pseudotransient continuation method and augment it with nonlinear inequality constraints. The constraint handling technique does not help in situations where no steady-state solution exists, for example due to an underresolved discretization of partial differential equations. However, an often overlooked situation is one in which the steady-state solution exists but cannot be reached by the solver, which typically fails due to the violation of constraints, i.e. a non-physical state error during state iterations. This is the shortcoming that we address by incorporating physical realizability constraints into the solution path from the initial condition to steady state. While we focus on the discontinuous Galerkin method applied to fluid dynamics, our technique relies only on implicit time marching and hence can be extended to other spatial discretizations and other physics problems. We analyze the sensitivity of the method to a range of input parameters and present results for compressible turbulent flows that show that the constrained method is significantly more robust than a standard unconstrained method while on par in terms of cost. Copyright © 0000 John Wiley & Sons, Ltd.

Smoothed particle hydrodynamics and its applications for multiphase flow and reactive transport in porous media

Smoothed particle hydrodynamics (SPH) is a Lagrangian method based on a meshless discretization of partial differential equations. In this review, we present SPH discretization of the Navier-Stokes and advection-diffusion-reaction equations, implementation of various boundary conditions, and time integration of the SPH equations, and we discuss applications of the SPH method for modeling pore-scale multiphase flows and reactive transport in porous and fractured media.

Invariant discretization of partial differential equations admitting infinite-dimensional symmetry groups

This paper is concerned with the invariant discretization of differential equations admitting infinite-dimensional symmetry groups. By way of example, we first show that there are differential equations with infinite-dimensional symmetry groups that do not admit enough joint invariants preventing the construction of invariant finite difference approximations. To solve this shortage of joint invariants we propose to discretize the pseudo-group action. Computer simulations indicate that the numerical schemes constructed from the joint invariants of discretized pseudo-group can produce better numerical results than standard schemes.

Analysis of the Monte-Carlo Error in a Hybrid Semi-Lagrangian Scheme

We consider Monte-Carlo discretizations of partial differential equations based on a combination of semi-lagrangian schemes and probabilistic representations of the solutions. We study the Monte-Carlo error in a simple case, and show that under an anti-CFL condition on the time-step $\delta t$ and on the mesh size $\delta x$ and for $N$ - the number of realizations - reasonably large, we control this error by a term of order $\mathcal{O}(\sqrt{\delta t /N})$. We also provide some numerical experiments to confirm the error estimate, and to expose some examples of equations which can be treated by the numerical method.

Generalized Summation-by-Parts Operators for the Second Derivative

The generalization of summation-by-parts operators for the first derivative of Del Rey Fernández, Boom, and Zingg [J. Comput. Phys., 266 (2014), pp. 214--239] is extended to approximations of second derivatives with a constant or variable coefficient. This enables the construction of second-derivative operators with one or more of the following characteristics: (i) nonrepeating interior point operators, (ii) nonuniform nodal distributions, and (iii) exclusion of one or both boundary nodes. Definitions are proposed that give rise to generalized summation-by-parts operators that result in consistent, conservative, and stable discretizations of partial differential equations with or without mixed derivatives. It is proven that approximations to the second derivative with a variable coefficient can be constructed using the constituent matrices of the constant-coefficient operator. Moreover, for operators with a repeating interior point operator, a decomposition is proposed that makes the application of such operators particularly straightforward. A number of novel operators are constructed, including operators on the Chebyshev--Gauss quadrature nodes and operators that have a repeating interior point operator but nonuniform nodal spacing near boundaries. The various operators are compared to the application of the first-derivative operator twice in the context of the linear convection-diffusion equation with a variable coefficient.

Non-Galerkin Multigrid Based on Sparsified Smoothed Aggregation

Algebraic multigrid (AMG) methods are known to be efficient in solving linear systems arising from the discretization of partial differential equations and other related problems. These methods employ a hierarchy of representations of the problem on successively coarser meshes. The coarse-grid operators are usually defined by (Petrov--)Galerkin coarsening, which is a projection of the original operator using the restriction and prolongation transfer operators. Therefore, these transfer operators determine the sparsity pattern and operator complexity of the multigrid hierarchy. In many scenarios the multigrid operators tend to become much denser as the coarsening progresses. Such behavior is especially problematic in parallel AMG computations, where it imposes an expensive communication overhead. In this work we present a new algebraic technique for controlling the sparsity pattern of the operators in the AMG hierarchy, independently of the choice of the restriction and prolongation. Our method is based on the aggregation multigrid framework, and it “sparsifies” smoothed aggregation operators while preserving their right and left near null-spaces. Numerical experiments for problems of convection-diffusion and diffusion with discontinuous coefficients demonstrate the efficiency and potential of this approach.

Preconditioning Techniques for Reduced Basis Methods for Parameterized Elliptic Partial Differential Equations

The reduced basis methodology is an efficient approach to solve parameterized discrete partial differential equations when the solution is needed at many parameter values. An offline step approximates the solution space, and an online step utilizes this approximation, the reduced basis, to solve a smaller reduced problem, which provides an accurate estimate of the solution. Traditionally, the reduced problem is solved using direct methods. However, the size of the reduced system needed to produce solutions of a given accuracy depends on the characteristics of the problem, and it may happen that the size is significantly smaller than that of the original discrete problem but large enough to make direct solution costly. In this scenario, it may be more effective to use iterative methods to solve the reduced problem. We construct preconditioners for reduced iterative methods which are derived from preconditioners for the full problem. This approach permits reduced basis methods to be practical for larger bases than direct methods allow. We illustrate the effectiveness of iterative methods for solving reduced problems by considering two examples, the steady-state diffusion and convection-diffusion-reaction equations.

Variational Integrators for Thermomechanical Coupled Dynamic Systems with Heat Conduction

AbstractVariational integrators are modern time‐integration schemes based on a discretization of the underlying variational principle. They thus skip the direct formulation and time discretization of partial differential equations. In mechanics, Hamilton's Principle is approximated by an action sum whose variation should be equal to zero, resulting in discrete Euler‐Lagrange Equations or equivalently in discrete Position‐Momentum Equations. Variational integrators are, by design, structure preserving (symplecticity) and show excellent long‐time behavior. In order to consider the coupling between mechanical and thermal quantities, Hamilton's principle is extended by using the notion of thermacy as thermal analogue to mechanical displacements. From this variational formulation, a variational integrator using the generalized trapezoidal rule is constructed exemplarily. A thermoelastic double pendulum with heat conduction serves as a model problem. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Model reduction for linear simulated moving bed chromatography systems using Krylov‐subspace methods

Simulated moving bed (SMB) chromatography is a well‐established technology for separating chemical compounds. To describe an SMB process, a finite‐dimensional multistage model arising from the discretization of partial differential equations is typically employed. However, its relatively high dimension poses severe computational challenges to various model‐based analysis. To overcome this challenge, two Krylov‐type model order reduction (MOR) methods are proposed to accelerate the computation of the cyclic steady states (CSSs) of SMB processes with linear isotherms. A “straightforward method” that carefully deals with the switching behavior in MOR is first proposed. Its improvement, a “subspace‐exploiting method,” thoroughly exploits each reduced model to achieve further acceleration. Simulation studies show that both methods achieve high accuracy and significant speedups. The subspace‐exploiting method turns out to be computationally much more efficient. Two challenging analyses of SMB processes, namely uncertainty quantification and CSS optimization, further demonstrate the accuracy, efficiency, and applicability of the proposed methods. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3773–3783, 2014

Distributed generic approximate sparse inverses

The need for accuracy in the solution of linear systems derived from the discretization of partial differential equations leads to large sparse linear systems. The solution of sparse linear systems requires efficient scalable methods. Iterative solvers require efficient parallel preconditioning methods to solve effectively sparse linear systems. Herewith, a new parallel algorithm for the generic approximate sparse inverse matrix method for distributed memory systems is proposed. The computation of the distributed generic approximate sparse inverse matrix is based on a column-wise approach, which allows the separation to independent problems that can be handled in parallel without synchronization points or intermediate communications. This is achieved by reforming the generic approximate sparse inverse matrix algorithm and its process of computation with a new partial solution method for the computation of the nonzero elements of each column dictated by the approximate inverse sparsity pattern. Moreover, an algorithmic scheme is proposed for the efficient distribution of data amongst the available workstations, along with a load balancing scheme for problems with large standard deviation in the number of nonzero elements per column. Numerical results are presented for the proposed schemes for various model problems.

Formula omitted]-Super-convergence of center finite difference method based on [formula omitted]-element for the elliptic equation

In this paper, the center finite difference (CFD) method for the elliptic equation is deduced by the P1-element and the first-order discrete partial differential equation over the dual element K∗ in the 1D or 2D domain. Next, the coefficient matrix of the CFD method is explicitly reduced and the H1-stability and convergence of the CFD solution uh is provided. Furthermore, the H1-super-convergence of uh to Ihu is obtained under the case of the almost-uniform mesh. Based on the H1-super-convergence of uh to Ihu, the optimal L2-error estimate of the numerical solution uh and the H1-super-convergence error estimate of the interpolation solution I2h2uh are derived respectively. Finally, some numerical tests are made to show the analytical results of the CFD method.

Analyzing the Vibration of Warp Yarns in Lengthwise and Crosswise

This article adopts nonlinear vibration method to analyze the fluctuation process of warp yarn. Selecting Kelvin model.A differential equation of the warp yarn vibration in lengthwise and crosswise is established by Newton law.And using Galerkin truncation method to separate the variables of time and space and to discrete partial differential equations into ordinary differential equations.The vertical and lateral vibration response of the warp are analyzed by 4-order Runge-Kutta method and matlab. Getting transverse and longitudinal vibration of warp yarns.

Research on Model and Virtual Simulation of Discrete Partial Differential Equation Based on Complex Statistical Perspective

This paper introduces the system structure of computer statistical perspective virtual simulation, and establishes an application database using parameter modification and optimization method. Then, a complex computer fluid solid coupling simulation model is established by using ANSYS software, and the mining results of computer streamline 3D data is obtained by the establishment of the fluid and solid domains data interface. At the same time, the computer virtual simulation system of statistical perspective is applied to the process of handball players training index data mining, and we will get the analysis curve of computer training resistance data mining. Finally, we establish the handball training intensity parameter storage library, to provide the technical reference for the handball athletes.