This book collects papers presented during the European Workshop on High Order Nonlinear Numerical Methods for Evolutionary PDEs (HONOM 2013) that was held at INRIA Bordeaux Sud-Ouest, Talence, France in March, 2013. The central topic is high order methods for compressible fluid dynamics. In the workshop, and in this proceedings, greater emphasis is placed on the numerical than the theoretical aspects of this scientific field. The range of topics is broad, extending through algorithm design, accuracy, large scale computing, complex geometries, discontinuous Galerkin, finite element methods, Lagrangian hydrodynamics, finite difference methods and applications and uncertainty quantification. These techniques find practical applications in such fields as fluid mechanics, magnetohydrodynamics, nonlinear solid mechanics, and others for which genuinely nonlinear methods are needed.

Methods for the numerical simulation of dynamic mathematical models have been the focus of intensive research for well over 60 years, and the demand for better and more efficient methods has grown as the range of applications has increased. Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in diverse applications such as fluid flow, image processing and computer vision, physics-based animation, mechanical systems, relativity, earth sciences, and mathematical finance. This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well. Audience: suitable for researchers and graduate students from a variety of fields including computer science, applied mathematics, physics, earth and ocean sciences, and various engineering disciplines. Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method.

"The goal of this thesis is to widen the class of provably convergent schemes for elliptic partial differential equations (PDEs) and improve their accuracy. We accomplish this by building on the theory of Barles and Souganidis, and its extension by Froese and Oberman to build monotone and filtered schemes.The first problem considered is the widely studied class of first order Hamilton-Jacobi (HJ) equations. The goal is to construct provably convergent accurate schemes, together with an efficient solver, by making use of the large number of discretizations and solvers already available. To this end, we build filtered schemes, whose main idea is to blend a stable monotone convergent scheme with an accurate scheme while retaining the advantages of both: stability and convergence of the former, and higher accuracy of the latter. Indeed, we are able to build schemes which are second, third, and fourth order accurate in one dimension, as well as schemes that are second order accurate in two dimensions. Moreover, the schemes are explicit, allowing us to use the easy-to-implement fast sweeping method. Using different accurate schemes (e.g. from standard centred differences, higher order upwinding and ENO interpolation), the accuracy of the filtered schemes is validated with computational results for the eikonal equation, as well as other HJ equations (both in one and two dimensions).The second problem considered is the 2-Hessian equation, a fully nonlinear PDE related to the intrinsic curvature for three-dimensional manifolds. The goal is to build numerical methods to compute its solution on a bounded domain given prescribed boundary data. We propose two distinct methods. The first is provably convergent to the unique viscosity solution. The second has higher accuracy and converges in practice, but lacks a formal proof of convergence. The PDE is elliptic on a restricted set of functions: a convexity-type constraint is needed for the ellipticity of the PDE operator, which poses additional difficulties when building the numerical methods. Solutions with both discretizations are obtained using Newton's method. Computational results are presented on a number of exact solutions which range in regularity from smooth to non-differentiable, and in shape from convex to non-convex.The third and last problem is to build a provably convergent scheme for the nonlinear PDE that governs the motion of level sets by affine curvature. It is closely related to mean curvature but exhibits instabilities not found in the former. These instabilities and the lack of regularity of the affine curvature operator posed unexpected and additional difficulties in building monotone schemes. A standard finite difference scheme is proposed and an example that illustrates its nonlinear instability is given. We build provably convergent monotone finite difference schemes. Numerical experiments demonstrate the accuracy and stability of the discretization, as well as the fact that our approximate solutions capture the affine invariance and morphological properties of the evolution." --

This book is open access under a CC BY 4.0 license. This easy-to-read book introduces the basics of solving partial differential equations by means of finite difference methods. Unlike many of the traditional academic works on the topic, this book was written for practitioners. Accordingly, it especially addresses: the construction of finite difference schemes, formulation and implementation of algorithms, verification of implementations, analyses of physical behavior as implied by the numerical solutions, and how to apply the methods and software to solve problems in the fields of physics and biology.

This book collects many of the presented papers, as plenary presentations, mini-symposia invited presentations, or contributed talks, from the European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) 2017. The conference was organized by the University of Bergen, Norway from September 25 to 29, 2017. Leading experts in the field presented the latest results and ideas in the designing, implementation, and analysis of numerical algorithms as well as their applications to relevant, societal problems. ENUMATH is a series of conferences held every two years to provide a forum for discussing basic aspects and new trends in numerical mathematics and scientific and industrial applications. These discussions are upheld at the highest level of international expertise. The first ENUMATH conference was held in Paris in 1995 with successive conferences being held at various locations across Europe, including Heidelberg (1997), Jyvaskyla (1999), lschia Porto (2001), Prague (2003), Santiago de Compostela (2005), Graz (2007), Uppsala (2009), Leicester (2011), Lausanne (2013), and Ankara (2015).

This proceedings volume gathers a selection of outstanding research papers presented at the third Conference on Isogeometric Analysis and Applications, held in Delft, The Netherlands, in April 2018. This conference series, previously held in Linz, Austria, in 2012 and Annweiler am Trifels, Germany, in 2014, has created an international forum for interaction between scientists and practitioners working in this rapidly developing field. Isogeometric analysis is a groundbreaking computational approach that aims to bridge the gap between numerical analysis and computational geometry modeling by integrating the finite element method and related numerical simulation techniques into the computer-aided design workflow, and vice versa. The methodology has matured over the last decade both in terms of our theoretical understanding, its mathematical foundation and the robustness and efficiency of its practical implementations. This development has enabled scientists and practitioners to tackle challenging new applications at the frontiers of research in science and engineering and attracted early adopters for this his novel computer-aided design and engineering technology in industry. The IGAA 2018 conference brought together experts on isogeometric analysis theory and application, share their insights into challenging industrial applications and to discuss the latest developments as well as the directions of future research and development that are required to make isogeometric analysis an established mainstream technology.

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest.