Published by De Gruyter

De Gruyter Series in Applied and Numerical Mathematics

Edited by: Rémi Abgrall, José Antonio Carrillo de la Plata, Jean-Michel Coron, Athanassios S. Fokas and Irene Fonseca

ISSN: 2512-1820

e-ISSN 2512-1847

Overview
Volumes

The series is devoted to the publication of high-level monographs and specialized graduate texts which cover the whole spectrum of applied mathematics, including its numerical aspects. The focus of the series is on the interplay between mathematical and numerical analysis, and also on its applications to mathematical models in the physical and life sciences.

The aim of the series is to be an active forum for the dissemination of up-to-date information in the form of authoritative works that will serve the applied mathematics community as the basis for further research.

Editorial Board

Rémi Abgrall, Universität Zürich, Switzerland
José Antonio Carrillo de la Plata, University of Oxford, UK
Jean-Michel Coron, Université Pierre et Marie Curie, Paris, France
Athanassios S. Fokas, Cambridge University, UK
Irene Fonseca, Carnegie Mellon University, Pittsburgh, USA

Book 2023

Finite Difference Methods for Nonlinear Evolution Equations

Zhi-Zhong Sun, Qifeng Zhang, Guang-hua Gao, China Science Publishing & Media Ltd.

Volume 8 in this series

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Nonlinear evolution equations are widely used to describe nonlinear phenomena in natural and social sciences. However, they are usually quite difficult to solve in most instances. This book introduces the finite difference methods for solving nonlinear evolution equations. The main numerical analysis tool is the energy method. This book covers the difference methods for the initial-boundary value problems of twelve nonlinear partial differential equations. They are Fisher equation, Burgers' equation, regularized long-wave equation, Korteweg-de Vries equation, Camassa-Holm equation, Schrödinger equation, Kuramoto-Tsuzuki equation, Zakharov equation, Ginzburg-Landau equation, Cahn-Hilliard equation, epitaxial growth model and phase field crystal model. This book is a monograph for the graduate students and science researchers majoring in computational mathematics and applied mathematics. It will be also useful to all researchers in related disciplines.

DOI:: https://doi.org/10.1515/9783110796018
ISBN:: 9783110796018
ISBN:: 9783110796117
ISBN:: 9783110795851
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

Book Ahead of Publication 2023

Metamaterial Analysis and Design

A Mathematical Treatment of Cochlea-inspired Sensors

Habib Ammari, Bryn Davies

Volume 9 in this series

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Metamaterials are advanced composite materials which have exotic and powerful properties. Their complicated microstructures make metamaterials challenging to model, requiring the use of sophisticated mathematical techniques. This book uses a from-first-principles approach (based on boundary integral methods and asymptotic analysis) to study a class of high-contrast metamaterials. These mathematical techniques are applied to the problem of designing graded metamaterials that replicate the function of the cochlea.

DOI:: https://doi.org/10.0000/9783110785135
ISBN:: 9783110785135
ISBN:: 9783110784961
ISBN:: 9783110784046
Subject:: Materials Sciences
Subject:: Materials Sciences, other
Subject:: Biomaterials and Natural Materials
Subject:: Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book Ahead of Publication 2024

Beyond Navier-Stokes

Advanced Computational Approaches for Rarefied Flows

Ramesh Agarwal, Rho-Shin Myong, Kun Xu, Wenwen Zhao

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This reference contains a comprehensive summary of recent methods beyond Navier-Stokes for computation of rarefied flows in continuum-transition regimes. Coverage includes descriptions and solutions for Burnett, and Grad’s moment equations, and description and solutions of classical and generalized Boltzmann equations.

DOI:: https://doi.org/10.0000/9783110747645
ISBN:: 9783110747645
ISBN:: 9783110747706
ISBN:: 9783110747539
Subject:: Engineering
Subject:: Mechanical Engineering
Subject:: Fluid mechanics
Subject:: Aeronautics & Astronautics
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2022

Numerical Simulation of Incompressible Viscous Flow

Methods and Applications

Roland Glowinski, Tsorng-Whay Pan

Volume 7 in this series

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This book on finite element-based computational methods for solving incompressible viscous fluid flow problems shows readers how to apply operator splitting techniques to decouple complicated computational fluid dynamics problems into a sequence of relatively simpler sub-problems at each time step, such as hemispherical cavity flow, cavity flow of an Oldroyd-B viscoelastic flow, and particle interaction in an Oldroyd-B type viscoelastic fluid. Efficient and robust numerical methods for solving those resulting simpler sub-problems are introduced and discussed. Interesting computational results are presented to show the capability of methodologies addressed in the book.

DOI:: https://doi.org/10.1515/9783110785012
ISBN:: 9783110785012
ISBN:: 9783110785050
ISBN:: 9783110784916
Subject:: Engineering
Subject:: Mechanical Engineering
Subject:: Fluid mechanics
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2022

Neural Networks and Numerical Analysis

Bruno Després

Volume 6 in this series

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This book uses numerical analysis as the main tool to investigate methods in machine learning and neural networks. The efficiency of neural network representations for general functions and for polynomial functions is studied in detail, together with an original description of the Latin hypercube method and of the ADAM algorithm for training. Furthermore, unique features include the use of Tensorflow for implementation session, and the description of on going research about the construction of new optimized numerical schemes.

DOI:: https://doi.org/10.1515/9783110783186
ISBN:: 9783110783186
ISBN:: 9783110783261
ISBN:: 9783110783124
Subject:: Computer Sciences
Subject:: Artificial Intelligence
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2020

Kinetic Equations

Volume 1: Boltzmann Equation, Maxwell Models, and Hydrodynamics beyond Navier–Stokes

Alexander V. Bobylev

Volume 5/1 in this series

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This two-volume monograph is a comprehensive and up-to-date presentation of the theory and applications of kinetic equations. The first volume covers many-particle dynamics, Maxwell models of the Boltzmann equation (including their exact and self-similar solutions), and hydrodynamic limits beyond the Navier-Stokes level.

DOI:: https://doi.org/10.1515/9783110550986
ISBN:: 9783110550986
ISBN:: 9783110550122
ISBN:: 9783110550177
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

Book 2019

Parabolic Equations with Irregular Data and Related Issues

Applications to Stochastic Differential Equations

Claude Le Bris, Pierre-Louis Lions

Volume 4 in this series

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This book studies the existence and uniqueness of solutions to parabolic-type equations with irregular coefficients and/or initial conditions. It elaborates on the DiPerna-Lions theory of renormalized solutions to linear transport equations and related equations, and also examines the connection between the results on the partial differential equation and the well-posedness of the underlying stochastic/ordinary differential equation.

DOI:: https://doi.org/10.1515/9783110635508
ISBN:: 9783110635508
ISBN:: 9783110633146
ISBN:: 9783110633139
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Publisher:: De Gruyter

Book 2018

Stochastically Forced Compressible Fluid Flows

Dominic Breit, Eduard Feireisl, Martina Hofmanová

Volume 3 in this series

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This book contains a first systematic study of compressible fluid flows subject to stochastic forcing. The bulk is the existence of dissipative martingale solutions to the stochastic compressible Navier-Stokes equations. These solutions are weak in the probabilistic sense as well as in the analytical sense. Moreover, the evolution of the energy can be controlled in terms of the initial energy. We analyze the behavior of solutions in short-time (where unique smooth solutions exists) as well as in the long term (existence of stationary solutions). Finally, we investigate the asymptotics with respect to several parameters of the model based on the energy inequality.

Contents
Part I: Preliminary results
Elements of functional analysis
Elements of stochastic analysis

Part II: Existence theory
Modeling fluid motion subject to random effects
Global existence
Local well-posedness
Relative energy inequality and weak–strong uniqueness

Part III: Applications
Stationary solutions
Singular limits

DOI:: https://doi.org/10.1515/9783110492552
ISBN:: 9783110492552
ISBN:: 9783110490763
ISBN:: 9783110490503
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Subject:: Mechanics and Fluid Dynamics
Publisher:: De Gruyter

Book 2017

Richardson Extrapolation

Practical Aspects and Applications

Zahari Zlatev, Ivan Dimov, István Faragó, Ágnes Havasi

Volume 2 in this series

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Scientists and engineers are mainly using Richardson extrapolation as a computational tool for increasing the accuracy of various numerical algorithms for the treatment of systems of ordinary and partial differential equations and for improving the computational efficiency of the solution process by the automatic variation of the time-stepsizes. A third issue, the stability of the computations, is very often the most important one and, therefore, it is the major topic studied in all chapters of this book.
Clear explanations and many examples make this text an easy-to-follow handbook for applied mathematicians, physicists and engineers working with scientific models based on differential equations.

Contents
The basic properties of Richardson extrapolation
Richardson extrapolation for explicit Runge-Kutta methods
Linear multistep and predictor-corrector methods
Richardson extrapolation for some implicit methods
Richardson extrapolation for splitting techniques
Richardson extrapolation for advection problems
Richardson extrapolation for some other problems
General conclusions

DOI:: https://doi.org/10.1515/9783110533002
ISBN:: 9783110533002
ISBN:: 9783110531985
ISBN:: 9783110516494
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

Book 2017

Free Boundaries in Rock Mechanics

Anvarbek Meirmanov, Oleg V. Galtsev, Reshat N. Zimin

Volume 1 in this series

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This monograph is concerned with free-boundary problems of partial differential equations arising in the physical sciences and in engineering. The existence and uniqueness of solutions to the Hele-Shaw problem are derived and techniques to deal with the Muskat problem are discussed. Based on these, mathematical models for the dynamics of cracks in underground rocks and in-situ leaching are developed.

Contents
Introduction
The Hele–Shaw problem
A joint motion of two immiscible viscous fluids
Mathematical models of in-situ leaching
Dynamics of cracks in rocks
Elements of continuum mechanics

DOI:: https://doi.org/10.1515/9783110546163
ISBN:: 9783110546163
ISBN:: 9783110545043
ISBN:: 9783110544909
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Subject:: Mechanics and Fluid Dynamics
Publisher:: De Gruyter