Blended block implicit methods for the numerical integration of
ODEs
Luigi Brugnano & Cecilia Magherini
(Dipartimento di Matematica ``U.Dini'',Università di Firenze, Italy)
We recall the basic facts concerning a new approach for naturally defining efficient nonlinear splittings for the numerical implementation of block implicit methods for ODEs. The approach, first derived in [1] and then further analyzed in [2], relies on the basic idea of defining a numerical method as the combination (blending) of two suitable component methods. By carefully choosing the methods, it is shown that very efficient implementations of block implicit methods can be obtained [3]. Moreover, some of them, characterized by a diagonal splitting, are well suited for parallel computers. A corresponding computational code [4] will be presented in a companion talk.
References
VMs): a Family
of Economical Implicit Methods for ODEs, Jour. Comput. Appl. Math. 116 (2000) 41-62.
The BiM code for the numerical integration of ODEs
Cecilia Magherini & Luigi Brugnano
(Dipartimento di Matematica ``U.Dini'',
Università di Firenze, Italy)
In this talk, numerical results obtained with a recently developed sequential code for ODEs [1,2], based on a blended implementation of Block Implicit Methods (BiM), are presented. The BiM code uses a variable order-variable stepsize strategy, with methods of order 4-6-8-10-12-14. Numerical tests from the CWI testset [3], comparing the new code with the most efficient existing ones, prove its effectiveness. The possibility of getting an improvement of the performance through a parallel implementation of the code, is also sketched.
References
TOM: A Code for Boundary Value Problems
Francesca Mazzia & Donato Trigiante
(Dipartimento Interuniversitario di Matematica, Bari, Italy)
Many choices need to be made in order to construct an efficient code for Boundary Value ODE Problems (BVP). The most crucial certainly are:
1. the discrete method;
2. the stepsize variation strategy;
3. the method for solving the nonlinear problems.
The existing codes, for example COLNEW, TWPBVP and MIRKDC [1,2, 6, 10], are based on different classes of methods such as collocation at gaussian points and mono-implicit Runge Kutta. The present code [7,8,9] is based on symmetric multistep formulas of highest order (TOMs) defined in [5].
Concerning the second choice, i.e. the mesh selection strategy, it is important especially when dealing with continuous problems whose solutions are of multiscale nature. Most of the existing codes base the mesh selection on an estimate of the local error. Such choice makes the code sensitive to the step size variation and, moreover, it does not provide any information about the conditioning of both continuous and discrete problem. We assume the principle that both the discrete and continuous problems should share the order of magnitude of the conditioning parameter.
In [3, 4] two quantities were defined which are related to the conditioning of the continuous problems and the corresponding discrete ones. Such parameters were also used to find a monitor function for the step size variation strategy.
Such step size variation has been refined and used in the present code [9]. It drastically reduces the computation cost, and moreover it also provide the correct mesh when we information about the behavior of the continuous solution is available.
Finally, for what concerns the the third major problem, i.e. the solution of the nonlinear problems, we use a quasilinearization strategy [8].
The effectiveness of choices made is shown by some numerical experiments on stiff problems. We also provide the corresponding results from COLNEW, TWPBVP and MIRKDC in order to show that the performance of the code is reasonable.
References
BVMs for the numerical approximation of BVPs arising in modeling of
Nonlinearly Elastic Materials
Ivonne Sgura & C. O. Horgan & G. Saccomandi
(Dipartimento di Matematica, Università di Lecce, Italy)
The purpose of this research is to investigate the mechanical
problem of the shearing of the annular region between two
concentric rigid cylinders occupied by an incompressible isotropic
nonlinearly elastic material. The deformation is driven by an
axial pressure gradient [4]. This kind of geometry, despite of its
simplicity, occurs in many engineering applications (for example
the rubber bush mountings in cab suspensions, etc...).
The problem is formulated as a two-point BVP for a second-order
nonlinear ODE. Two classes of neo-Hookean materials are considered
[3,5]. The first class models limiting chain extensibility at the
molecular level and the second class is of power-law type, that
includes modeling of biological materials.
Our purpose is to investigate the behavior of
the solutions for different values of the pressure gradient in
order to predict the possible breaking points in the material.
The numerical experiments are performed with the
experimental code TOM [6].
The code uses a class of BVM [2], with
the quasi-linearization
technique [7] and a mesh selection
strategy based on the conditioning of the problem [1,8].
We provide numerical results regarding softening
power-law materials and we show that a boundary layer
behavior is exhibited near the bonded surfaces. We perform further
numerical experiments for hardening materials showing that
an interior localization is exhibited, that is a cusp
occurs in the axial displacement profile when large pressure
gradients are considered. The numerical results highlight the
contrasting behavior between softening and hardening rubber-like
or biological materials.
References