Adaptive Time-Stepping and Computational Stability

A glass ceramic sintered product obtained by sintering a mixture powder containing a BaO-containing glass, metal oxide particles having a coefficient of linear thermal expansion at 40 to 400° C. of not smaller than 6 ppm/°C., and a Zr compound and, particularly, containing the Zr compound in an amount of from 0.1 to 30% by weight calculated as Zr02, and exhibiting a coefficient of linear thermal expansion at 40 to 400° C. of from 8.5 to 18 ppm/°C. The glass ceramic sintered product exhibits excellent resistance against chemicals and does not discolor even when subjected to treatment with an acidic or alkaline solution in the step of plating. A wiring board using the glass ceramic sintered product as an insulating substrate exhibits a coefficient of linear thermal expansion substantially similar to that of an external circuit board, effectively suppressing the occurrence of thermal stress and cracking.

[1]  Christoph Uberhuber.  Numerical Computation, Volume I , 1997 .

[2]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[3]  R. Skeel.  Construction of variable-stepsize multistep formulas , 1986 .

[4]  C. Führer,et al.  A collocation formulation of multistep methods for variable step-size extensions , 2002 .

[5]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[6]  Kjell Gustafsson,et al.  Control Strategies for the Iterative Solution of Nonlinear Equations in ODE Solvers , 1997, SIAM J. Sci. Comput..

[7]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[8]  Gustaf Söderlind,et al.  Automatic Control and Adaptive Time-Stepping , 2002, Numerical Algorithms.

[9]  Karl Johan Åström,et al.  Computer-Controlled Systems: Theory and Design , 1984 .

[10]  Kjell Gustafsson,et al.  Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods , 1991, TOMS.

[11]  J. M. Watt.  Numerical Initial Value Problems in Ordinary Differential Equations , 1972 .

[12]  Christoph W. Ueberhuber.  Numerical computation : methods, software, and analysis , 1997 .

[13]  Gustaf Söderlind,et al.  Digital filters in adaptive time-stepping , 2003, TOMS.

[14]  Christoph W. Ueberhuber.  Numerical Computation 1 , 1997 .

[15]  W. M. Lioen,et al.  Test set for IVP solvers , 1996 .

[16]  Astrom.  Computer Controlled Systems , 1990 .

[17]  Tom Tietenberg,et al.  Theory and design , 2001 .

[18]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[19]  Kjell Gustafsson,et al.  Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods , 1991, TOMS.