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Solving Ordinary Differential Equations II: Stiff and Differential - Algebraic Problems

TL;DR: In this paper, the authors present the solution of stiff differential equations and differential-algebraic systems (differential equations with constraints) and discuss their application in physics, chemistry, biology, control engineering, electrical network analysis, and computer programs.
Abstract: The subject of this book is the solution of stiff differential equations and of differential-algebraic systems (differential equations with constraints). The book is divided into four chapters. The beginning of each chapter is of introductory nature, followed by practical applications, the discussion of numerical results, theoretical investigations on the order and accuracy, linear and nonlinear stability, convergence and asymptotic expansions. Stiff and differential-algebraic problems arise everywhere in scientific computations (e.g., in physics, chemistry, biology, control engineering, electrical network analysis, mechanical systems). Many applications as well as computer programs are presented.
Citations
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Journal ArticleDOI
TL;DR: Modules for Experiments in Stellar Astrophysics (MESA) as discussed by the authors is an open source software package for modeling the evolution of stellar structures and composition. But it is not suitable for large-scale systems such as supernovae.
Abstract: We substantially update the capabilities of the open source software package Modules for Experiments in Stellar Astrophysics (MESA), and its one-dimensional stellar evolution module, MESA star. Improvements in MESA star's ability to model the evolution of giant planets now extends its applicability down to masses as low as one-tenth that of Jupiter. The dramatic improvement in asteroseismology enabled by the space-based Kepler and CoRoT missions motivates our full coupling of the ADIPLS adiabatic pulsation code with MESA star. This also motivates a numerical recasting of the Ledoux criterion that is more easily implemented when many nuclei are present at non-negligible abundances. This impacts the way in which MESA star calculates semi-convective and thermohaline mixing. We exhibit the evolution of 3-8 M ? stars through the end of core He burning, the onset of He thermal pulses, and arrival on the white dwarf cooling sequence. We implement diffusion of angular momentum and chemical abundances that enable calculations of rotating-star models, which we compare thoroughly with earlier work. We introduce a new treatment of radiation-dominated envelopes that allows the uninterrupted evolution of massive stars to core collapse. This enables the generation of new sets of supernovae, long gamma-ray burst, and pair-instability progenitor models. We substantially modify the way in which MESA star solves the fully coupled stellar structure and composition equations, and we show how this has improved the scaling of MESA's calculational speed on multi-core processors. Updates to the modules for equation of state, opacity, nuclear reaction rates, and atmospheric boundary conditions are also provided. We describe the MESA Software Development Kit that packages all the required components needed to form a unified, maintained, and well-validated build environment for MESA. We also highlight a few tools developed by the community for rapid visualization of MESA star results.

2,761 citations

Journal ArticleDOI
TL;DR: The current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness, are described.
Abstract: SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.

2,124 citations

Posted Content
TL;DR: The SUNDIALS suite of nonlinear and DIfferential/ALgebraic equation solvers (SUNDIALs) as mentioned in this paper has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures.
Abstract: In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout this work, we have adhered to specific guiding principles that minimized the impact to current users while providing maximum flexibility for later evolution of solvers and data structures. The redesign was done through creation of new classes for linear and nonlinear solvers, enhancements to the vector class, and the creation of modern Fortran interfaces that leverage interoperability features of the Fortran 2003 standard. The vast majority of this work has been performed "behind-the-scenes," with minimal changes to the user interface and no reduction in solver capabilities or performance. However, these changes now allow advanced users to create highly customized solvers that exploit their problem structure, enabling SUNDIALS use on extreme-scale, heterogeneous computational architectures.

1,858 citations

Journal ArticleDOI
TL;DR: In this paper, a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles is presented, including the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge-Kutta schemes.
Abstract: This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented.

1,657 citations

Book
07 Aug 2013
TL;DR: In this paper, an Adams-type predictor-corrector method for the numerical solution of fractional differential equations is discussed, which may be used both for linear and nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator) too.
Abstract: We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.

1,617 citations