Scalar Partial Differential Equations Research Articles

AMG based on compatible weighted matching for GPUs

We describe main issues and design principles of an efficient implementation, tailored to recent generations of Nvidia Graphics Processing Units (GPUs), of an Algebraic MultiGrid (AMG) preconditioner previously proposed by one of the authors and already available in the open-source package BootCMatch: Bootstrap algebraic multigrid based on Compatible weighted Matching for standard CPUs. The AMG method relies on a new approach for coarsening sparse symmetric positive definite (s.p.d.) matrices, named coarsening based on compatible weighted matching. It exploits maximum weight matching in the adjacency graph of the sparse matrix, driven by the principle of compatible relaxation, providing a suitable aggregation of unknowns which goes beyond the limits of the usual heuristics applied in the current methods. We adopt an approximate solution of the maximum weight matching problem, based on a recently proposed parallel algorithm, referred to as the Suitor algorithm, and show that it allows us to obtain good quality coarse matrices for our AMG on GPUs. We exploit inherent parallelism of modern GPUs in all the kernels involving sparse matrix computations both for the setup of the preconditioner and for its application in a Krylov solver, outperforming preconditioners available in the original sequential CPU code as well as the single node Nvidia AmgX library. Results for a large set of linear systems arising from discretization of scalar and vector partial differential equations (PDEs) are discussed.

Improving solve time of aggregation‐based adaptive AMG

SummaryThis paper proposes improving the solve time of a bootstrap algebraic multigrid (AMG) designed previously by the authors. This is achieved by incorporating the information, a set of algebraically smooth vectors, generated by the bootstrap algorithm, in a single hierarchy by using sufficiently large aggregates, and these aggregates are compositions of aggregates already built throughout the bootstrap algorithm. The modified AMG method has good convergence properties and shows significant reduction in both memory and solve time. These savings with respect to the original bootstrap AMG are illustrated on some difficult (for standard AMG) linear systems arising from discretization of scalar and vector function elliptic partial differential equations in both 2D and 3D.

On integration of the system of MHD equations modeling wave processes in a rotating liquid with arbitrary magnetic Reynolds number

The paper is concerned with the dynamics of large-scale wave processes in a rotating layer of inviscid conducting incompressible liquid of variable depth. The problem is modelled as a system of partial differential equations with necessary boundary conditions. With the help of auxiliary functions, the above system of partial differential magnetohydrodynamic equations is reduced to a single scalar partial differential equation. An exact analytic solution of the small perturbation problem is obtained. It is shown that if the external magnetic field is parallel to the axis of rotation of the layer, then the magnetic field decays for finite values of the magnetic Reynolds number.

Boundary observer design for a class of semi-linear hyperbolic PDE systems with recycle loop

ABSTRACT The present manuscript considers the observer design problem for a class of scalar semi-linear hyperbolic partial differential equation (PDE) systems with a recycle loop through the boundary point. The design method of Kazantzis and Kravaris [(1998). Nonlinear observer design using Lyapunov's auxiliary theorem. Systems & Control Letters, 34(5), 241–247] developed for the nonlinear finite dimensional systems observer design is extended to semi-linear hyperbolic PDE systems. The observer design problem is tackled through a first-order associated PDE. Due to the existence of spatial partial derivative operator, Lyapunov's Auxiliary Theorem originated from finite-dimensional systems can be no longer applied to seek conditions ensuring solvability of the associated PDE. In this manuscript, a new theorem is formulated and proved to ensure the solvability. The solution for the associated PDE is locally analytic nonlinear coordinate transformation which provides foundation for observer realization and a series solution approach is developed. Simulation examples are presented to study the performance of the proposed observer.

Compact high-order numerical schemes for scalar hyperbolic partial differential equations

Abstract This work is a comparative study on the numerical behavior of compact high-order schemes for discretizing scalar hyperbolic problems. To this aim, we investigate two possibilities: a high-order Hermitian scheme (HUPS), based upon the scalar variable and its first derivatives in space, and a “classical” high-order upwind scheme (CUPS), based upon the scalar variable and a wide numerical stencil, as a reference scheme. To study the theoretical properties of both schemes we use a spectral analysis based on a Fourier transform of the numerical solution. In a one-dimensional context, this analysis enables us to investigate the impact of the “spurious” components generated by the Hermitian schemes. As demonstrated by the spectral analysis, in some circumstances, this “spurious” component may deeply alter the accuracy of the scheme and even, in some cases, destroy its consistency. Thus, by identifying the salient features of a Hermitian scheme, we try to design efficients Hermitian schemes for multi-dimensional purposes. A two-dimensional Fourier analysis and some scalar numerical tests allow to highlight two high-order versions of a Hermitian scheme (HUPS4,5) of which the numerical properties are compared with a classical fifth-order scheme (CUPS5) based upon a wider stencil.

Boundary stabilization of a class of reaction–advection–diffusion systems via a gradient-based optimization approach

In this paper, the boundary stabilization problem of a class of unstable reaction–advection–diffusion (RAD) systems described by a scalar parabolic partial differential equation (PDE) is considered. Different the previous research, we present a new gradient-based optimization framework for designing the optimal feedback kernel for stabilizing the unstable PDE system. Our new method does not require solving non-standard Riccati-type or Klein–Gorden-type PDEs. Instead, the feedback kernel is parameterized as a second-order polynomial whose coefficients are decision variables to be tuned via gradient-based dynamic optimization, where the gradients of the system cost functional (which penalizes both kernel and output magnitude) with respect to the decision parameters are computed by solving a so-called “costate” PDE in standard form. Special constraints are imposed on the kernel coefficients to ensure that the optimized kernel yields closed-loop stability. Finally, three numerical examples are illustrated to verify the effectiveness of the proposed approach.

Fuzzy Control With Guaranteed Cost for Nonlinear Coupled Parabolic PDE-ODE Systems via PDE Static Output Feedback and ODE State Feedback

This paper investigates the guaranteed cost fuzzy control (GCFC) problem for a class of nonlinear systems modeled by an $n$ -dimension ordinary differential equation (ODE) coupled with a semilinear scalar parabolic partial differential equation (PDE). A Takagi–Sugeno (T–S) fuzzy coupled parabolic PDE-ODE model is initially proposed to accurately represent the nonlinear coupled system. Then, on the basis of the T–S fuzzy coupled model, a GCFC design is developed in terms of linear matrix inequalities to exponentially stabilize the coupled system while providing an upper bound for a prescribed quadratic cost function. The proposed fuzzy control scheme consists of the ODE state feedback and the PDE static output feedback employing locally collocated piecewise uniform actuators and sensors. Moreover, a suboptimal GCFC problem is also addressed to minimize the cost bound. Finally, the developed method is applied to the cruise control and surface temperature cooling of a hypersonic rocket car.

Multigoal-oriented error estimates for non-linear problems

Abstract In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton’s method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.

Direct error in constitutive equation formulation for inverse heat conduction problem

SummaryWe present a mixed numerical formulation that handles discontinuities well for scalar hyperbolic partial differential equations. The formulation is based on a least‐square error in the constitutive equation. It is motivated by scalar inverse diffusion problems with interior data and applies to convection of a passive scalar in a discontinuous compressible flow field. We motivate the need for a mixed formulation by discretizing using an irreducible finite element method and discuss some of the limitations of that approach. We then develop and prove that the mixed formulation is well posed and verify that it works for problems with continuous and discontinuous thermal conductivity distributions.

On a L∞ functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity

In this paper, we consider a L∞ functional derivative estimate for the first spatial derivatives of bounded classical solutions u:RN×[0,T]→R to the Cauchy problem for scalar second order semi-linear parabolic partial differential equations with a continuous nonlinearity f:R→R and initial data u0:RN→R, of the form,maxi=1,…,N⁡(supx∈RN⁡|uxi(x,t)|)≤Ft(f,u0,u)∀t∈[0,T]. Here Ft:At→R is a functional as defined in §1 and x=(x1,x2,…,xn)∈RN. We establish that the functional derivative estimate is non-trivially sharp, by constructing a sequence (fn,0,u(n)), where for each n∈N, u(n):RN×[0,T]→R is a solution to the Cauchy problem with zero initial data and nonlinearity fn:R→R, and for which there exists α>0 such thatmaxi=1,…,N⁡(supx∈RN⁡|uxi(n)(x,T)|)≥α, whilstlimn→∞⁡(inft∈[0,T]⁡(maxi=1,…,N⁡(supx∈RN⁡|uxi(n)(x,t)|)−Ft(fn,0,u(n))))=0.

Optimal control of coupled parabolic–hyperbolic non-autonomous PDEs: infinite-dimensional state-space approach

ABSTRACTThis paper deals with the design of an optimal state-feedback linear-quadratic (LQ) controller for a system of coupled parabolic–hypebolic non-autonomous partial differential equations (PDEs). The infinite-dimensional state space representation and the corresponding operator Riccati differential equation are used to solve the control problem. Dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the LQ-optimal control problem and also to guarantee the exponential stability of the closed-loop system. Thanks to the eigenvalues and eigenfunctions of the parabolic operator and also the fact that the hyperbolic-associated operator Riccati differential equation can be converted to a scalar Riccati PDE, an algorithm to solve the LQ control problem has been presented. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ optimal controller designed in the early portion of the paper is implemented for the original non-linear model. Numerical simulations are performed to show the controller performances.

A Stencil Scaling Approach for Accelerating Matrix-Free Finite Element Implementations

We present a novel approach to fast on-the-fly low order finite element assembly for scalar elliptic partial differential equations of Darcy type with variable coefficients optimized for matrix-free implementations. Our approach introduces a new operator that is obtained by appropriately scaling the reference stiffness matrix from the constant coefficient case. Assuming sufficient regularity, an a priori analysis shows that solutions obtained by this approach are unique and have asymptotically optimal order convergence in the $H^1$- and the $L^2$-norms on hierarchical hybrid grids. For the preasymptotic regime, we present a local modification that guarantees uniform ellipticity of the operator. Cost considerations show that our novel approach requires roughly one-third of the floating-point operations compared to a classical finite element assembly scheme employing nodal integration. Our theoretical considerations are illustrated by numerical tests that confirm the expectations with respect to accuracy and run-time. A large scale application with more than a hundred billion ($1.6\cdot10^{11}$) degrees of freedom executed on 14310 compute cores demonstrates the efficiency of the new scaling approach.

PI-control design of continuum models of production systems governed by scalar hyperbolic partial differential equation

A production system which produces plenty of items in many steps can be modelled as a continuous flow problem governed by a nonlinear and nonlocal hyperbolic partial differential equation. One way to adjust the output of such process is by manipulating the start rate. This paper considers the control and regulation by proportional-integral (PI) controllers for the continuum production systems. In the considered system, the input and output are located at the boundary of production system. In particular, the closed-loop stability of the linearized continuum production model with the designed PI-controller is proved using spectral analysis and Lyapunov theory. Numerical results demonstrate successful tracking for step inputs in the demand rate.

Relationships between symmetries depending on arbitrary functions and integrals of discrete equations

The paper is devoted to the conjecture that an equation is Darboux integrable if and only if it possesses symmetries that depend on arbitrary functions. We note that the results of previous works together prove this conjecture for scalar partial differential equations of the form . For autonomous semi-discrete and discrete analogues of these equations, we prove that the sequence of Laplace invariants is terminated by zero for an equation if this equation admits an operator mapping any function of one independent variable into a symmetry of the equation. The vanishing of a Laplace invariant allows us to construct a formal integral, i.e. an operator that maps symmetries into integrals (including, generally speaking, trivial integrals). This paper and the results of previous works together prove a ‘formal’ version of the aforementioned conjecture for semi-discrete and pure discrete cases.

Generalizations of the short pulse equation.

We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.

Design of Integral Controllers for Nonlinear Systems Governed by Scalar Hyperbolic Partial Differential Equations

The paper deals with the control and regulation by integral controllers forthe nonlinear systems governed by scalar quasi-linear hyperbolic partial differentialequations. Both the control input and the measured output are located on the boundary.The closed-loop stabilization of the linearized model with the designed integral controlleris proved first by using the method of spectral analysis and then by the Lyapunov directmethod. Based on the elaborated Lyapunov function we prove local exponential stabilityof the nonlinear closed-loop system with the same controller. The output regulationto the set-point with zero static error by the integral controller is shown uponthe nonlinear system. Numerical simulations by the Preissmann scheme are carriedout to validate the robustness performance of the designed controllerto face to unknown constant disturbances.

Application of Quasi-Steady-State Methods to Nonlinear Models of Intracellular Transport by Molecular Motors.

Molecular motors such as kinesin and dynein are responsible for transporting material along microtubule networks in cells. In many contexts, motor dynamics can be modelled by a system of reaction-advection-diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the full PDE system. Here, we extend this QSS reduction methodology to certain nonlinear reaction models. The QSS method relies on the assumption that the nonlinear binding and unbinding interactions of the cellular motors occur on a faster timescale than the spatial diffusion and advection processes. The full system dynamics are shown to be well approximated by the dynamics on the slow manifold. The slow manifold is parametrized by a single scalar quantity that satisfies a scalar nonlinear PDE, called the QSS PDE. We apply the QSS method to several specific nonlinear models for the binding and unbinding of molecular motors, and we use the resulting approximations to draw conclusions regarding the parameter dependence of the spatial distribution of motors for these models.

Local exponential stabilization via boundary feedback controllers for a class of unstable semi-linear parabolic distributed parameter processes

This paper addresses the problem of local exponential stabilization via boundary feedback controllers for a class of nonlinear distributed parameter processes described by a scalar semi-linear parabolic partial differential equation (PDE). Both the domain-averaged measurement form and the boundary measurement form are considered. For the boundary measurement form, the collocated boundary measurement case and the non-collocated boundary measurement case are studied, respectively. For both domain-averaged measurement case and collocated boundary measurement case, a static output feedback controller is constructed. An observer-based output feedback controller is constructed for the non-collocated boundary measurement case. It is shown by the contraction semigroup theory and the Lyapunov’s direct method that the resulting closed-loop system has a unique classical solution and is locally exponentially stable under sufficient conditions given in term of linear matrix inequalities (LMIs). The estimation of domain of attraction is also discussed for the resulting closed-loop system in this paper. Finally, the effectiveness of the proposed control methods is illustrated by a numerical example.

Data-assimilated computational fluid dynamics modeling of convection-diffusion-reaction problems

This study focuses on the development and application of a data-assimilated multi-algorithm which combines a fourth-order finite-volume computational fluid dynamics (CFD) algorithm and the ensemble Kalman filter (EnKF) data assimilation algorithm for solving the problems involving important physics relevant to wave convection, molecular diffusion, and reaction of interest to engineering science. This data-assimilated CFD algorithm is applied to estimate an uncertain model parameter. By restricting the problems to one- and two-dimensional spatial space, this study allows us to develop a greater understanding of the integrated algorithm without the complexity or computational cost associated with three dimensions. Although scalar partial differential equations are used, fundamental issues in the algorithm development and application of EnKF to the physical processes occurring in a domain on engineering scales are sufficiently illuminated. Results of the one-dimensional convection-diffusion-reaction problem and the two-dimensional flame propagation demonstrate the validity of the data-assimilated CFD modeling system.

Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part II

In this two-part paper, a high-order accurate implicit mesh discontinuous Galerkin (dG) framework is developed for fluid interface dynamics, facilitating precise computation of interfacial fluid flow in evolving geometries. The framework uses implicitly defined meshes—wherein a reference quadtree or octree grid is combined with an implicit representation of evolving interfaces and moving domain boundaries—and allows physically prescribed interfacial jump conditions to be imposed or captured with high-order accuracy. Part one discusses the design of the framework, including: (i) high-order quadrature for implicitly defined elements and faces; (ii) high-order accurate discretisation of scalar and vector-valued elliptic partial differential equations with interfacial jumps in ellipticity coefficient, leading to optimal-order accuracy in the maximum norm and discrete linear systems that are symmetric positive (semi)definite; (iii) the design of incompressible fluid flow projection operators, which except for the influence of small penalty parameters, are discretely idempotent; and (iv) the design of geometric multigrid methods for elliptic interface problems on implicitly defined meshes and their use as preconditioners for the conjugate gradient method. Also discussed is a variety of aspects relating to moving interfaces, including: (v) dG discretisations of the level set method on implicitly defined meshes; (vi) transferring state between evolving implicit meshes; (vii) preserving mesh topology to accurately compute temporal derivatives; (viii) high-order accurate reinitialisation of level set functions; and (ix) the integration of adaptive mesh refinement.In part two, several applications of the implicit mesh dG framework in two and three dimensions are presented, including examples of single phase flow in nontrivial geometry, surface tension-driven two phase flow with phase-dependent fluid density and viscosity, rigid body fluid–structure interaction, and free surface flow. A class of techniques known as interfacial gauge methods is adopted to solve the corresponding incompressible Navier–Stokes equations, which, compared to archetypical projection methods, have a weaker coupling between fluid velocity, pressure, and interface position, and allow high-order accurate numerical methods to be developed more easily. Convergence analyses conducted throughout the work demonstrate high-order accuracy in the maximum norm for all of the applications considered; for example, fourth-order spatial accuracy in fluid velocity, pressure, and interface location is demonstrated for surface tension-driven two phase flow in 2D and 3D. Specific application examples include: vortex shedding in nontrivial geometry, capillary wave dynamics revealing fine-scale flow features, falling rigid bodies tumbling in unsteady flow, and free surface flow over a submersed obstacle, as well as high Reynolds number soap bubble oscillation dynamics and vortex shedding induced by a type of Plateau–Rayleigh instability in water ripple free surface flow. These last two examples compare numerical results with experimental data and serve as an additional means of validation; they also reveal physical phenomena not visible in the experiments, highlight how small-scale interfacial features develop and affect macroscopic dynamics, and demonstrate the wide range of spatial scales often at play in interfacial fluid flow.