Abstract: Introduction 1. Linear Algebra Part I. Mathematical Stability and Ill Conditioning. 2. Systems of Linear Algebraic Equations 3. 4. Differential and Difference Equations Part II. Discretization Error 5. Discretization Error for Initial Problems 6. Discretization Error for Boundary Value Problems Part III. Convergence of Iterative Methods 7. Systems of Linear Equations 8. Systems of Nonlinear Equations Part IV. Rounding Error 9. Rounding Error for Gaussian Elimination Bibliography Index.

Abstract: In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

Abstract: We introduce new results connecting differential and morphological operators that provide a formal and theoretically grounded approach for stable and fast contour evolution. Contour evolution algorithms have been extensively used for boundary detection and tracking in computer vision. The standard solution based on partial differential equations and level-sets requires the use of numerical methods of integration that are costly computationally and may have stability issues. We present a morphological approach to contour evolution based on a new curvature morphological operator valid for surfaces of any dimension. We approximate the numerical solution of the curve evolution PDE by the successive application of a set of morphological operators defined on a binary level-set and with equivalent infinitesimal behavior. These operators are very fast, do not suffer numerical stability issues, and do not degrade the level set function, so there is no need to reinitialize it. Moreover, their implementation is much easier since they do not require the use of sophisticated numerical algorithms. We validate the approach providing a morphological implementation of the geodesic active contours, the active contours without borders, and turbopixels. In the experiments conducted, the morphological implementations converge to solutions equivalent to those achieved by traditional numerical solutions, but with significant gains in simplicity, speed, and stability.

Abstract: In this paper, a phase-field-based multiple-relaxation-time lattice Boltzmann (LB) model is proposed for incompressible multiphase flow systems. In this model, one distribution function is used to solve the Chan-Hilliard equation and the other is adopted to solve the Navier-Stokes equations. Unlike previous phase-field-based LB models, a proper source term is incorporated in the interfacial evolution equation such that the Chan-Hilliard equation can be derived exactly and also a pressure distribution is designed to recover the correct hydrodynamic equations. Furthermore, the pressure and velocity fields can be calculated explicitly. A series of numerical tests, including Zalesak's disk rotation, a single vortex, a deformation field, and a static droplet, have been performed to test the accuracy and stability of the present model. The results show that, compared with the previous models, the present model is more stable and achieves an overall improvement in the accuracy of the capturing interface. In addition, compared to the single-relaxation-time LB model, the present model can effectively reduce the spurious velocity and fluctuation of the kinetic energy. Finally, as an application, the Rayleigh-Taylor instability at high Reynolds numbers is investigated.

Abstract: This paper describes a new formulation of the material point method (MPM) for solving coupled hydromechanical problems of fluid-saturated soil subjected to large deformation. A soil-pore fluid coupled MPM algorithm based on Biot's mixture theory is proposed for solving hydromechanical interaction problems that include changes in water table location with time. The accuracy of the proposed method is examined by comparing the results of the simulation of a one-dimensional consolidation test with the corresponding analytical solution. A sensitivity analysis of the MPM parameters used in the proposed method is carried out for examining the effect of the number of particles per mesh and mesh size on solution accuracy. For demonstrating the capability of the proposed method, a physical model experiment of a large-scale levee failure by seepage is simulated. The behavior of the levee model with time-dependent changes in water table matches well to the experimental observations. The mechanisms of seepage-induced failure are discussed by examining the pore-water pressures, as well as the effective stresses computed from the simulations.

Abstract: The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a nonlocal operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solutions with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.

Abstract: The three-dimensional nonlinear size-dependent motion characteristics of a microbeam are investigated numerically, with special consideration to one-to-one internal resonances between the in-plane and out-of-plane transverse modes. All of the in-plane and out-of-plane displacements and inertia are taken into account and Hamilton’s principle, in conjunction with the modified couple stress theory, is employed to obtain the nonlinear partial differential equations governing the motions of the system in the in-plane and out-of-plane directions. The discretization procedure is carried out by applying the Galerkin technique to the partial differential equations of motion, yielding a set of nonlinear ordinary differential equations. A linear analysis is performed upon this set of equations so as to obtain the size-dependent natural frequencies of the system. The nonlinear analysis of the discretized equations of motion is carried out by employing the pseudo-arclength continuation technique, resulting in the resonant responses of the system. It is shown that, due to the presence of one-to-one internal resonances between the in-plane and out-of-plane transverse modes, an in-plane excitation can give rise to an out-of-plane displacement; the internal resonances also cause the occurrence of extra solution branches and new bifurcation points.

Abstract: Many methods are available for the solution of radiative heat transfer problems in participating media. Among these, the discrete ordinates method (DOM) and the finite volume method (FVM) are among the most widely used ones. They provide a good compromise between accuracy and computational requirements, and they are relatively easy to integrate in CFD codes. This paper surveys recent advances on these numerical methods. Developments concerning the grid structure (e.g., new formulations for axisymmetrical geometries, body-fitted structured and unstructured meshes, embedded boundaries, multi-block grids, local grid refinement), the spatial discretization scheme, and the angular discretization scheme are described. Progress related to the solution accuracy, solution algorithm, alternative formulations, such as the modified DOM and FVM, even-parity formulation, discrete-ordinates interpolation method and method of lines, and parallelization strategies is addressed. The application to non-gray media, variable refractive index media, and transient problems is also reviewed.

Abstract: In this paper, a consistent meshfree Lagrangian approach for numerical analysis of incompressible flow with free surfaces, named least squares moving particle semi-implicit (LSMPS) method, is developed. The present methodology includes arbitrary high-order accurate meshfree spatial discretization schemes, consistent time integration schemes, and generalized treatment of boundary conditions. LSMPS method can resolve the existing major issues of widely used strong-form particle method for incompressible flow—particularly, the lack of consistency condition for spatial discretization schemes, difficulty in enforcing consistent Neumann boundary conditions, and serious instability like unphysical pressure oscillation. Applications of the present proposal demonstrate remarkable enhancements of stability and accuracy.

Abstract: Different numerical methods have been implemented to simulate internal natural convection heat transfer and also to identify the most accurate and efficient one. A laterally heated square enclosure, filled with air, was studied. A FORTRAN code based on the lattice Boltzmann method (LBM) was developed for this purpose. The finite difference method was applied to discretize the LBM equations. Furthermore, for comparison purpose, the commercially available CFD package FLUENT, which uses finite volume Method (FVM), was also used to simulate the same problem. Different discretization schemes, being the first order upwind, second order upwind, power law, and QUICK, were used with the finite volume solver where the SIMPLE and SIMPLEC algorithms linked the velocity-pressure terms. The results were also compared with existing experimental and numerical data. It was observed that the finite volume method requires less CPU usage time and yields more accurate results compared to the LBM. It has been noted that the 1st order upwind/SIMPLEC combination converges comparatively quickly with a very high accuracy especially at the boundaries. Interestingly, all variants of FVM discretization/pressure-velocity linking methods lead to almost the same number of iterations to converge but higher-order schemes ask for longer iterations.

Abstract: We give an algorithmic introduction to Lagrangian coherent structures (LCSs) using a newly developed computational engine, LCS Tool. LCSs are most repelling, attracting and shearing material lines that form the centerpieces of observed tracer patterns in two-dimensional unsteady dynamical systems. LCS Tool implements the latest geodesic theory of LCSs for two-dimensional flows, uncovering key transport barriers in unsteady flow velocity data as explicit solutions of differential equations. After a review of the underlying theory, we explain the steps and numerical methods used by LCS Tool, and illustrate its capabilities on three unsteady fluid flow examples.

Abstract: We propose an $hp$-version discontinuous Galerkin finite element method for fully nonlinear second-order elliptic Hamilton--Jacobi--Bellman equations with Cordes coefficients. The method is proved to be consistent and stable, with convergence rates that are optimal with respect to mesh size, and suboptimal in the polynomial degree by only half an order. Numerical experiments on problems with nonsmooth solutions and strongly anisotropic diffusion coefficients illustrate the accuracy and computational efficiency of the scheme. An existence and uniqueness result for strong solutions of the fully nonlinear problem and a semismoothness result for the nonlinear operator are also provided.