An efficient hybrid numerical method for the high-order Allen–Cahn equation

In this thesis, we study systems of nonlinear partial differential equations that can be used to describe phase separation. This process can be modeled from both a macroscopic and microscopic point of view. Typically, the macroscopic model contains a local differential operator, which accounts for short-range interactions. In the microscopic model, however, this operator is then replaced by a nonlocal operator, which describes long-range interactions between the particles. This is done by convolution integrals weighted by suitable interaction kernels. In the first part, we analyze the asymptotic behavior of the nonlocal operator in the case, when the corresponding interaction kernel concentrates around the origin suitably. We prove that the nonlocal operator converges to a local differential operator as the parameter in the interaction kernel is sent to zero. More precisely, we even provide concrete rates of convergence depending on this parameter. The analysis is done in the case of sufficiently smooth bounded domains and a large class of interaction kernels. Indeed, the kernel can be either anisotropic or isotropic as well as of $W^{1,1}$-regularity or even more singular. The convergence is shown with respect to the $L^p$-norm, where $p\in[1,\infty)$. The proof is based on localization and perturbation arguments. In the case of regular kernels, we also prove convergence on the torus with respect to the $L^2$-norm. This proof is based on methods from Fourier analysis. In the second part, we focus on models, where two different phases are separated by a diffuse interface. We intend to apply the results from the first part to show that solutions to the nonlocal model converge to a solution to the local model as the parameter in the nonlocal operator is sent to zero. More precisely, we prove strong convergence for the sequence of solutions along with certain rates of convergence. This is done using an energy method. We consider the following models: - Cahn-Hilliard equation - Allen-Cahn equation - Cahn-Hilliard model for tumor growth - Cahn-Hilliard/Navier-Stokes system Finally, we also investigate the case, where both the parameter in the nonlocal operator and the thickness of the diffuse interface are sent to zero suitably. The analysis is done for the nonlocal Allen--Cahn equation on the torus. We first prove a well-posedness result and suitable error estimates for the solutions. Then, we combine the convergence result from the first part together with existing results for the sharp interface limit of the local Allen--Cahn equation.

Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo’s fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of time-fractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo’s time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research.

Prediction of radiative heat transfer in 2D irregular geometries using the collocation spectral method based on body-fitted coordinates

We propose in this paper an efficient and accurate numerical method for the spectral fractional Laplacian equation using the Caffarelli–Silvestre extension. In particular, we propose several strategies to deal with the singularity and the additional dimension associated with the extension problem: (i) reducing the $$d+1$$ dimensional problem to a sequence of d-dimension Poisson-type problems by using the matrix diagonalizational method; (ii) resolving the singularity by applying the enriched spectral method in the extended dimension. We carry out rigorous analysis for the proposed numerical method, and provide abundant numerical examples to verify the theoretical results and illustrate effectiveness of the proposed method.

In this chapter, the stability of decentralized control systems is investigated, which is shown to be dependent on the notion of decentralized fixed modes (DFMs). Different techniques are provided to characterize the DFMs of a general proper finite-dimensional LTI system. The first technique is an efficient numerical method which can be easily utilized on MATLAB. It basically uses the random number generator to check if any of the eigenvalues of the system remains fixed w.r.t. a generic constant gain matrix of the desired decentralized structure. The other techniques are analytical and need to verify the rank of a set of matrices obtained from the system matrices. One of the techniques requires checking the transmission zeros of a set of systems. The technique is initially given for a decentralized control system with SISO subsystems and is then extended to the systems with MIMO subsystems. The main characteristic of this method is that it requires the transmission zero of every (virtual) system in a set to be checked, which implies that the ranks of all of the corresponding matrices need to be verified. The other method, on the other hand, requires the ranks of a different set of matrices to be checked. The main difference of this method with the former one is that if any of the rank conditions does not hold, the corresponding eigenvalue is a DFM and there is no need to check the remaining matrices. Furthermore, a numerical method is provided which unlike the one described earlier does not require the choice of generic gain matrices. Finally, a procedure is introduced to find a LTI decentralized controller of proper structure to locate the poles of the closed-loop system in desired locations in the complex plane. The procedure uses centralized design techniques to find the local controllers.

Oscillators are key components of electronic systems. In RF communication systems, they are used for frequency translation of information signals and for channel selection, and in digital electronic systems, they are used as a time reference, i.e. a clock signal, in order to synchronize operations. Undesired perturbations in practical electronic systems adversely affect the spectral and timing properties of oscillators, which is a key performance limiting factor, being a major contributor to bit-error-rate (BER) of RF communication systems, and creating synchronization problems in clocked and sampled-data systems. Characterizing how perturbations affect oscillators is therefore crucial for practical applications. The traditional approach to analysing perturbed nonlinear systems (i.e. linearization) is not valid for oscillators. In this paper, we present a theory and efficient numerical methods, for non-linear perturbation and noise analysis of oscillators described by a system of differential-algebraic equations (DAEs). Our techniques can be used in characterizing phase noise and timing jitter due to intrinsic noise in IC devices, and evaluating the effect of substrate and supply noise on the timing properties of practical oscillators. In this paper, we also establish novel results for periodically time-varying systems of linear DAEs, which we rely on in developing the above theory and the numerical methods. Copyright © 2000 John Wiley & Sons, Ltd.

Long-time simulations with complex code using multiple nodes of Intel Xeon Phi Knights Landing

Lagrangian generic second order traffic flow models for node

When a plane electromagnetic wave impinges on a diffraction grating or other periodic structures, reflected and transmitted waves propagate away from the structure in different radiation channels. A diffraction anomaly occurs when the outgoing waves in one or more radiation channels vanish. Zero reflection, zero transmission, and perfect absorption are important examples of diffraction anomalies, and they are useful for manipulating electromagnetic waves and light. Since diffraction anomalies appear only at specific frequencies and/or wave vectors, and may require the tuning of structural or material parameters, they are relatively difficult to find by standard numerical methods. Iterative methods may be used, but good initial guesses are required. To determine all diffraction anomalies in a given frequency interval, it is necessary to repeatedly solve the diffraction problem for many frequencies. In this paper, an efficient numerical method is developed for computing diffraction anomalies. The method relies on nonlinear eigenvalue formulations for scattering anomalies and solves the nonlinear eigenvalue problems by a contour-integral method. Numerical examples involving periodic arrays of cylinders are presented to illustrate the new method.

It is shown that many classical and singular value loop shaping problems are closed-loop convex. Consequently, loop shaping problems can be solved by efficient numerical methods. In particular, it can be determined whether or not a compensator exists that satisfies a given set of loop shaping specifications. Loop shaping design problems that are formulated as classical optimization problems, e.g., maximizing bandwidth subject to given margin and cutoff specifications, can be solved by direct numerical methods for quasiconvex optimization. A consequence of these observations is that closed-loop convex design methods can be used to do compensator design in a classical loop shaping framework which is familiar to many control engineers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In this paper, an efficient numerical method is proposed to handle two-dimensional fractional diffusion equations on a finite domain. The proposed method combines the product of Legendre wavelet bases for two spatial dimensions and a time direction. The operational matrix of the proposed method is obtained. Tikhonov regularization is employed to stabilize the system in cases where the final linear system of equations is large. The convergence analysis of the method is studied and some numerical examples are presented to investigate the efficiency and accuracy of the method.

The boundary element method (BEM) is an efficient numerical method that approximates solutions of various boundary value problems. Despite its success little research has been performed on the conditioning of the linear systems that appear in the BEM. For a Laplace equation with Dirichlet boundary conditions a remarkable phenomenon is observed; the corresponding boundary integral equation (BIE) is singular for a certain critical size of the 2D domain. As a consequence the discrete counterpart of the BIE, the linear system, is singular too, or at least ill-conditioned. This is reflected by the condition number of the system matrix, which is infinitely large, or at least very large. When the condition number of the BEM-matrix is large, the linear system is difficult to solve and the solution of the system is very sensible to perturbations in the boundary data. For a Laplace equation with mixed boundary conditions a similar phenomenon is observed. The corresponding BEM-matrix consists of two blocks; one block originates from the BEM-matrix belonging to the Dirichlet problem, the other block originates from the BEM-matrix belonging to the Neumann problem. The composite matrix inherits the solvability problems from the Dirichlet block. In other words, for the Laplace equation with mixed boundary conditions there exists also a critical size of the 2D domain for which the BEM-matrix has an infinitely large condition number. Hence the size and shape of the domain affects the solvability of the BEM problem. The critical size of the domain for which the BIE becomes singular is related to the logarithmic capacity of the domain. The logarithmic capacity is a positive real number that is a function of the size and shape of the domain. If this logarithmic capacity is equal to one, the domain is a critical domain, and for this domain the BIE becomes singular. Thus by computing the logarithmic capacity we can a-priori determine whether the BIE will be singular or not. The logarithmic capacity depends linearly on the scale of the domain, and thus a domain with logarithmic capacity equal to one can always be found by rescaling the domain. Unfortunately the logarithmic capacity can only be computed analytically for a few simple domains; for more involved domains the logarithmic capacity can be estimated though. There are several possibilities to avoid large condition numbers, i.e. singular BIEs that appear at critical domains. The first option is to rescale the domain such that the logarithmic capacity is unequal to one. One can also add a supplementary condition to the BIE and the linear system. A drawback of this option is that the linear system has more equations than unknowns and different techniques are required to solve the system. A third option is to slightly modify the fundamental solution of the Laplace operator. This fundamental solution directly appears in the BIE and it can be shown that a suitable modification yields BIEs that do not become singular. The critical domains for which the BIEs become singular do not restrict to Laplace equations only. Also for BIEs applied to the biharmonic equation or the elastostatic equations and the Stokes equations such critical domains exist. As the last two equations are vectorial equations, also the corresponding BIE consists of two equations. As a consequence two critical domains can be found for which these BIEs become singular. To obtain nonsingular BIEs techniques similar to the Laplace case can be used. Unfortunately we cannot a-priori determine the sizes for which the BIEs becomes singular, and thus do not know to what size we should rescale the domain to obtain nonsingular BIEs. The existence of critical domains is in essence caused by the logarithmic term in the fundamental solutions for the elliptic boundary value problems in 2D. This logarithmic term does not depend linearly on the size of the domain. When a domain is scaled, i.e. multiplied by a scale factor, the argument of the logarithm is also multiplied by this scale factor, but the logarithm turns this into an additive term. Thus the logarithm transforms multiplication into addition. This affects the BIEs in such a way that critical domains can appear. The fundamental solutions of boundary value problems in 3D do not contain a logarithmic term. Hence scaling of the domain does not affect the fundamental solution, and consequently also the BIE is not affected. Hence we may safely rescale 3D domains without the risk to encounter a critical domain. An example in which a domain takes many different sizes and shapes is the blowing problem. In this problem a viscous fluid is blown to a desired shape. Typically the time is discretised into a set of discrete time steps, and at each step the shape of the fluid is computed by solving the Stokes equations. When attempting to simulate this problem in 2D, we meet a large number of 2D domains, and we risk that one of these domains is equal to or approaches a critical domain. In such a case the BEM will have difficulties with solving the Stokes equations for that particular domain. When simulating the blowing problem in 3D, no critical domains are encountered. It turns out that the BEM is a very efficient numerical method for this particular 3D problem with a moving boundary. As we are merely interested in the shape of the fluid, we only need to know the flow of its boundary. The BEM does exactly that; it does not compute the flow at the interior of the fluid. Furthermore it is rather easy to include other effects from the blowing problem in the model, such as gravity, surface tension and friction from the contact of the fluid with a wall. As only the boundary of the fluid is discretised, the system matrices that appear in the BEM are smaller than the system matrices that appear when solving the problem with a finite element method, for example. Though the BEM-matrices are dense, while the finite element matrices are sparse, the computational effort for the BEM is relatively low. In short, the BEMis a very appropriate numerical method when solving blowing problems.

Diffusion-weighted magnetic resonance imaging is an advanced imaging technique that provides detailed information for clinical diagnosis and treatment monitoring. The purpose of this thesis is to establish fractional dynamic models for better characterising anomalous diffusion in biological tissues of the human brain and explore reliable numerical methods. The introduced generalised fractional operators and variable diffusion coefficients provide more possibilities for capturing structural heterogeneity and diffusion complexities in biological tissues. Robust and efficient numerical methods are developed to solve the proposed fractional systems with lower computational costs.

Fluid-Structure Interaction (FSI) problems represent complex coupled systems that are critical in engineering applications such as aerospace, biomechanics, and civil structures. Efficient numerical methods are vital for optimizing these problems due to the nonlinear and multiphysics nature of FSI systems. This article reviews and evaluates several state-of-the-art numerical optimization techniques used in FSI, including finite element methods (FEM), finite volume methods (FVM), and monolithic versus partitioned solvers. Emphasis is placed on algorithmic efficiency, convergence behavior, and real-world case study applications. A comparative analysis is presented to assess performance and robustness of the discussed methods.

A fast numerical method for the valuation of American lookback put options