Journal ArticleDOI
Embedded dirk methods for the numerical integration of stiff systems of odes
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TLDR
In this article, the optimal order of embedded pairs of Diagonally Implicit Runge-Kutta (DIRK) methods is examined, and an analysis of A and L-stability properties of q-stage order q DIRK methods with unequal diagonal elements is presented.Abstract:
In this paper, the optimal order of embedded pairs of Diagonally Implicit Runge-Kutta (DIRK) methods is examined. It is shown that a q-stage DIRK method of order p embedded in a q + 1 stage DIRK method of order p + 1 cannot have p = q + 1. Thus adopting embedding techniques to estimate the local truncation error results in giving up an order of accuracy for q<6. Embedded pairs of orders two and three for the basic method are derived with the additional stage being either explicit or implicit. Numerical results indicate that significant savings are realized when the extra stage is explicit. An analysis of A and L-stability properties of q-stage order q DIRK methods with unequal diagonal elements is presented. Necessary and sufficient conditions for A and L-stability are derived. To assess the potential of such methods, a number of embedded DIRK formulas are implemented. Numerical results for selected test problems are presented.read more
Citations
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Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review
TL;DR: A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken and ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances.
Journal ArticleDOI
A void growth‐based failure model to describe spallation
TL;DR: In this article, a dynamic failure model to describe void nucleation, growth, and coalescence in ductile metals is reported based on a pressure-dependent yield criterion for compressible plastic flow.
High strain rate behavior of metals, ceramics, and concrete
TL;DR: In this article, the authors investigated the high strain rate behavior of various metals and ceramics through a combined experimental and computational approach, including quasi-static, split Hopkinson bar, rod-on rod, plate-on-rod, and plate impact test configurations.
Journal ArticleDOI
A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows
Alexander Jaust,Jochen Schütz +1 more
TL;DR: Numerical results demonstrate the performance of the hybridized discontinuous Galerkin (HDG) method for both linear and nonlinear (systems of) time-dependent convection-diffusion equations.
Journal ArticleDOI
Implicit Multiderivative Collocation Solvers for Linear Partial Differential Equations with Discontinuous Galerkin Spatial Discretizations
TL;DR: This work constructs novel discretizations for the unsteady convection–diffusion equation that include two-point collocation methods with multiple time derivatives as well as a sixth-order fully implicit collocation method that only requires a total of three stages.
References
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Journal ArticleDOI
Implicit Runge-Kutta processes
TL;DR: In this article, it is assumed that the Taylor expansions for y and y may be terminated at any term with an error of the same order as the first term omitted, and all derivatives exist and are continuous.
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Diagonally Implicit Runge–Kutta Methods for Stiff O.D.E.’s
TL;DR: In this paper, it was shown that no 4-stage method of this type has order 5, and that it is impossible for a strongly S-stable diagonally implicit method to attain order 4 in 4 stages.
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COMPARING NUMERICAL METHODS FOR STIFF SYSTEMS OF O.D.E :s
TL;DR: In this article, the authors describe a technique for comparing numerical methods that have been designed to solve stiff systems of ordinary differential equations, and the basis of a fair comparison is discussed in detail.
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A special family of Runge-Kutta methods for solving stiff differential equations
TL;DR: A family of methods of Implicit Runge-Kutta Methods is constructed and some results concerning their maximum attainable order and stability properties are given.
Journal ArticleDOI
An implementation of singly-implicit Runge-Kutta methods
Kevin Burrage,Kevin Burrage,Kevin Burrage,John C. Butcher,John C. Butcher,John C. Butcher,F. H. Chipman,F. H. Chipman,F. H. Chipman +8 more
TL;DR: An overview of the structure of the algorithm is provided together with a general description of how it is used, and an Algol 60 procedure declaration forSTRIDE is included together with the listing of an equivalent Fortran subroutine.