Exponential integrators for second-order in time partial differential equations

Two types of second-order in time partial differential equations (PDEs), namely semilinear wave equations and semilinear beam equations are considered. To solve these equations with exponential integrators, we present an approach to compute efficiently the action of the matrix exponential as well as those of related matrix functions. Various numerical simulations are presented that illustrate this approach.Comment: 19 pages, 10 figure

Delay differential equations in a nonlinear cochlear model.

The human auditory system performs its primary function in the cochlea, the main organ of the inner ear, where the spectral analysis of a sound signal and its transduction into a neural signal occur. It is filled with liquid and divided in two cavities by the basilar membrane (BM). A sound stimulus propagates in air as an acoustic pressure wave through the outer and the middle ear. The pressure of the stapes on the oval window (boundary between the middle and the inner ear) causes the cochlear fluid to flow between the two cavities through a hole at the end of the BM. A spatial partial differential equation of fluid-dynamics describes this physical process. As a consequence of the differential pressure between the two cavities, each micro-element of the BM oscillates as a forced damped harmonic oscillator. The BM displacement is amplified by the overlying outer hair cells (OHCs) through a nonlinear nonlocal active feedback mechanism. The latter can be modeled by means of various representations. Among them, the delayed stiffness model of Talmadge et al. (J. Acoust. Soc. Am. 104, 1998) has been considered in this thesis. Specifically, the cochlear nonlinearity is introduced as a quadratic function of the BM displacement in the passive linear damping function. Moreover, the active mechanism is described by two additional forces, each one proportional to the BM displacement delayed by a slow and a fast feedback constant time, respectively. According to this model, a time delay differential equation (DDE) of the second order describes the oscillating dynamics of the BM. A different formulation of the nonlinear active mechanism, driven by the OHCs, is expressed as a nonlinear function of the BM velocity by the anti-damping model of Moleti et al. (J. Acoust. Soc. Am. 133, 2013). In this case the model equations do not contain time delays. The numerical integration of the above mentioned models has been obtained by finite differencing with respect to the space variable in the state space, as introduced by Elliott et al. (J. Acoust. Soc. Am. 122, 2007), and then integrating in time with the adaptive package introduced by Bertaccini and Sisto as a modification of the popular Matlab ode15s package (J. Comput. Phys. 230, 2011). The semidiscrete formulation of the delayed stiffness model and the anti-damping model has a non trivial mass matrix, and eigenvalues of the system matrix with large negative real part and imaginary part. That is why an implicit solver with an infinite region of absolute stability should be used. Therefore, the customized Matlab ode15s package by Bertaccini and Sisto seems to be the convenient choice to integrate the problem at hand numerically. In particular, for the delayed stiffness model, an integrator for constant DDEs (the method of steps; Bellen and Zennaro, Oxford University Press 2003) has been formulated and based on the customized ode15s. All these topics have been discussed in this doctoral thesis, which is subdivided in the following chapters. Chapter 1 describes the anatomy of the human ear, with special regard to the cochlea. Some experimental evidences about the cochlear mechanisms are discussed, in order to support the cochlear modeling. Two physical models with one degree of freedom are shown: the anti-damping model of Sisto et al. (J. Acoust. Soc. Am. 128, 2010) and Moleti et al. (J. Acoust. Soc. Am. 133, 2013), and the delayed stiffness model of Talmadge et al. (J. Acoust. Soc. Am. 104, 1998). Chapter 2 discusses the general theory of DDEs, with greater reference to constant and time dependent DDEs from Bellen and Zennaro (Oxford University Press 2003). Existence and uniqueness of time dependent DDEs are briefly analyzed, while the method of steps is shown as a basic approach to find a numerical approximation of the DDEs solution. According to this method, IVPs of constant DDEs (as for the semidiscrete delayed stiffness model) are turned into IVPs of ODEs in a subinterval (of length less than or equal to the time delay) of the whole integration interval. Each IVP of ODEs can be integrated by means of any ODEs numerical method, and its convergence is then discussed. Chapter 3 describes the main tools used to find an approximate solution of the considered models. In particular, the discretization for spatial partial derivatives by means of finite differences is shown. Such a representation turns a model, which is continuous in the space-time domain, into a semidiscrete model to be integrated in time. The models considered in this thesis are stiff, so the phenomenon of stiffness is discussed and the ode15s package of Matlab for integrating stiff ODEs is described. Nevertheless, greater benefits can be obtained by using the ode15s package customized by Bertaccini and Sisto as a hybrid direct-iterative solver which exploits Krylov subspace methods. Chapter 4 shows the semidiscrete formulation of the continuous models (anti-damping model and delayed stiffness model) in the state space with respect to the spatial variable, as introduced by Elliott et al. (J. Acoust. Soc. Am. 122, 2007). The algebraic properties of the semidiscrete models are discussed in order to show why the customized ode15s package may perform a faster numerical integration of the semidiscrete models and how this solver can be used in an integration numerical technique for constant DDEs (the method of steps). Chapter 5 shows the results produced by the numerical experiments of the delayed stiffness model by supplying a sinusoidal tone, and compares them with the numerical results produced by the anti-damping model. Some considerations about the numerical approach of the time integration are also discussed, and a part of the simplified code used for integrating the semidiscrete delayed stiffness model, is reported. The results are comparable with those obtained by the anti-damping model, and then the numerical experimental evidences seem to justify the proposed integration technique for constant DDEs. Delayed model properties of tonotopicity, anti-damping and nonlinearity are verified, as well as the dependence of the approximate solution on some free parameters of the model. The cochlear response described by the delayed stiffness model shows a typical tall and broad BM activity pattern. This behavior is also found in the numerical results of a model with two degree of freedom produced by Neely and Kim (J. Acoust. Soc. Am. 79, 1986) and Elliott et al. (J. Acoust. Soc. Am. 122, 2007)

Energetic BEM for the numerical analysis of 2D Dirichlet damped wave propagation exterior problems

Abstract Time-dependent problems modeled by hyperbolic partial differential equations can be reformulated in terms of boundary integral equations and solved via the boundary element method. In this context, the analysis of damping phenomena that occur in many physics and engineering problems is a novelty. Starting from a recently developed energetic space-time weak formulation for 1D damped wave propagation problems rewritten in terms of boundary integral equations, we develop here an extension of the so-called energetic boundary element method for the 2D case. Several numerical benchmarks, whose numerical results confirm accuracy and stability of the proposed technique, already proved for the numerical treatment of undamped wave propagation problems in several dimensions and for the 1D damped case, are illustrated and discussed

Runge-Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations

Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency. However, as computers become ever more parallel, horizontally explicit solutions of the equations of atmospheric motion might become an attractive alternative due to the additional inter-processor communication of implicit methods. Implicit and explicit (IMEX) time-stepping schemes have long been combined in models of the atmosphere using semi-implicit, split-explicit or HEVI splitting. However, most studies of the accuracy and stability of IMEX schemes have been limited to the parabolic case of advection–diffusion equations. We demonstrate how a number of Runge–Kutta IMEX schemes can be used to solve hyperbolic wave equations either semi-implicitly or HEVI. A new form of HEVI splitting is proposed, UfPreb, which dramatically improves accuracy and stability of simulations of gravity waves in stratified flow. As a consequence it is found that there are HEVI schemes that do not lose accuracy in comparison to semi-implicit ones. The stability limits of a number of variations of trapezoidal implicit and some Runge–Kutta IMEX schemes are found and the schemes are tested on two vertical slice cases using the compressible Boussinesq equations split into various combinations of implicit and explicit terms. Some of the Runge–Kutta schemes are found to be beneficial over trapezoidal, especially since they damp high frequencies without dropping to first-order accuracy. We test schemes that are not formally accurate for stiff systems but in stiff limits (nearly incompressible) and find that they can perform well. The scheme ARK2(2,3,2) performs the best in the tests

Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part

In this paper, we extend the implicit-explicit (IMEX) methods of Peer type recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203--215, 2017] to a broader class of two-step methods that allow the construction of super-convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes combine the necessary stability of implicit and low computational costs of explicit methods to efficiently solve systems of ordinary differential equations with both stiff and non-stiff parts included in the source term. To construct super-convergent IMEX-Peer methods with favourable stability properties, we derive necessary and sufficient conditions on the coefficient matrices and apply an extrapolation approach based on already computed stage values. Optimised super-convergent IMEX-Peer methods of order s+1 for s=2,3,4 stages are given as result of a search algorithm carefully designed to balance the size of the stability regions and the extrapolation errors. Numerical experiments and a comparison to other IMEX-Peer methods are included.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with arXiv:1610.0051

Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation

Numerical methods for simulation of electrical activity in the myocardial tissue

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) and partial differential equations (PDEs) that describe the behaviour of this electrical activity. Generating an efficient numerical solution of these models is a challenging task, and in fact the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this thesis, we discuss two sets of experiments that test ideas for making the numerical solution process more efficient. In the first set of experiments, we examine the numerical solution of four single cell cardiac electrophysiological models, which consist solely of ODEs. We study the efficiency of using implicit-explicit Runge-Kutta (IMEX-RK) splitting methods to solve these models. We find that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform most methods commonly used in practice for two of the models, and they outperform all methods commonly used in practice for the remaining models. In the second set of experiments, we examine the solution of the bidomain model, a model consisting of both ODEs and PDEs that are typically solved separately. We focus these experiments on numerical methods for the solution of the two PDEs in the bidomain model. The most popular method for this task, the Crank-Nicolson method, produces unphysical oscillations; we propose a method based on a second-order L-stable singly diagonally implicit Runge-Kutta (SDIRK) method to eliminate these oscillations. We find that although the SDIRK method is able to eliminate these unphysical oscillations, it is only more efficient for crude error tolerances