Implicit Explicit Methods For Evolutionary Partial Differential Equations
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Implicit-Explicit Methods for Evolutionary Partial Differential Equations
Implicit-explicit (IMEX) time discretization methods have proven to be highly effective for the numerical solution of a wide class of evolutionary partial differential equations (PDEs) across various contexts. These methods have become mainstream for solving evolutionary PDEs, particularly in the fields of hyperbolic and kinetic equations. The first book on the subject, Implicit-Explicit Methods for Evolutionary Partial Differential Equations provides an in-depth yet accessible approach. The authors summarize and illustrate the construction, analysis, and application of IMEX methods using examples, test cases, and implementation details; guide readers through the various methods and teach them how to select and use the one most appropriate for their needs; and demonstrate how to identify stiff terms and effectively implement high-order methods in time for a variety of systems of PDEs. Readers interested in learning modern techniques for the effective numerical solution of evolutionary PDEs with multiple time scales will find in this book a unified, compact, and accessible treatment. This book is intended for applied mathematicians, scientists, and engineers who use or are interested in learning about IMEX schemes. Readers should have some background in numerical methods for ODE systems and basic finite difference and finite volume discretization of evolutionary PDEs, along with a basic understanding of the relevant mathematical models. The book is suitable for students who have had a basic course in numerical analysis and are familiar with partial differential equations.
Numerical Methods for Evolutionary Differential Equations
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.
Computational Mechanics and Applied Mathematics: Perspectives from Young Scholars
This book collects the latest advances and innovations in the field of applied mathematics and computational mechanics, as presented at the 2nd Workshop GIMC SIMAI YOUNG, held in Naples, Italy, on July 10–12, 2024. The workshop was the joint effort of Computational Mechanics Group of the Italian Association of Theoretical and Applied Mechanics -AIMETA (GIMC) and Italian Society of Applied and Industrial Mathematics (SIMAI) and was meant to highlight the works of young researchers in the field. Topics include mathematical models for socio-epidemiological dynamics, efficient numerical methods for evolutionary PDEs, multi-scale approaches and machine learning techniques in material modelling, nonlinear material behaviour, computational methods for shells and spatial structures, assessment, monitoring, and design of masonry structures, particles in numerical simulations, non-Newtonian complex fluids, mathematical modelling in mechanobiology and oncology, mechanics of biological systems and bioinspired materials, computational approaches for complex dynamical systems, optimization methods for classical and data-driven approaches. The contributions, which were selected by means of a rigorous peer-review process, present a wealth of exciting ideas that will open novel research directions and foster multidisciplinary collaboration.