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USCD1226 LSOIBT.

LSOIBT, Implicit Ordinary Differential Equations System Block Tridiagonal Matrices

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1. NAME OR DESIGNATION OF PROGRAM:  LSOIBT.
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2. COMPUTERS
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Program name Package id Status Status date
LSOIBT USCD1226/01 Tested 23-SEP-2005

Machines used:

Package ID Orig. computer Test computer
USCD1226/01 IBM PC PC Windows
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3. DESCRIPTION OF PROGRAM OR FUNCTION

LSOIBT is a set of general-purpose FORTRAN routines solver for the initial value problem for ordinary differential equation systems. It is suitable for both stiff and nonstiff systems. LSOIBT treat systems in the linearly implicit form A(t,y) dy/dt = g(t,y), A = a square matrix, i.e. with the derivative dy/dt implicit, but linearly so. It allows A to be singular, in which case the system is a differential-algebraic equation (DAE) system.  In that case, the user must be very careful to supply a well-posed problem with consistent initial conditions. LSOIBT, written jointly with C. S. Kenney, solves linearly implicit systems in which the matrices involved are all assumed to be block-tridiagonal.  Linear systems are solved by the LU method. The LSOIBT source is commented extensively to facilitate modification. Both a single-precision version and a double-precision version are available.
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4. METHODS

This program solves a semi-discretized form of the following system of three PDEs (each similar to a Burgers equation):
u(i)   =  -(u(1)+u(2)+u(3)) u(i)   +  eta(i) u(i)    (i=1,2,3),
     t                           x                xx
on the interval  -1 .le. x .le. 1, and with time t .ge. 0.
The diffusion coefficients are eta(*) = .1, .02, .01.
The boundary conditions are u(i) = 0 at x = -1 and x = 1 for all i.
The initial profile for each u(i) is a square wave:
     u(i) = 0         on 1/2 .lt. abs(x) .le. 1
     u(i) = amp(i)/2  on abs(x) = 1/2
     u(i) = amp(i)    on 0 .le. abs(x) .lt. 1/2
where the amplitudes are amp(*) = .2, .3, .5.
A simplified Galerkin treatment of the spatial variable x is used, with piecewise linear basis functions on a uniform mesh of 100 intervals.  The result is a system of ODEs in the discrete values u(i,k) approximating u(i)  (i=1,2,3) at the interior points (k = 1,...,99).  
The ODEs are:
       (u'(i,k-1) + 4 u'(i,k) + u'(i,k+1))/6  =
         -(1/6dx) (c(k-1)dul(i) + 2c(k)(dul(i)+dur(i)) + c(k+1)dur(i))
         + (eta(i)/dx**2) (dur(i) - dul(i))     (i=1,2,3,  k=1,...,99),
where
         c(j) = u(1,j)+u(2,j)+u(3,j),   dx = .02 = the interval size,
         dul(i) = u(i,k) - u(i,k-1),   dur(i) = u(i,k+1) - u(i,k).
Terms involving boundary values (subscripts 0 or 100) are dropped from the equations for k = 1 and k = 99 above.
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5. RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM
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6. TYPICAL RUNNING TIME
USCD1226/01
At the NEA-DB the demonstration program included in this package ran on a PC Windows Xeon in a few seconds.
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7. UNUSUAL FEATURES
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8. RELATED OR AUXILIARY PROGRAMS

This program is part of the ODEPACK (USCD1232) collection of Fortran solvers for the initial value problem for ordinary differential equation systems.  It consists of nine solvers, namely a basic solver called LSODE (USCD1223) and eight variants of it: LSODES (USCD1229), LSODA (USCD1227), LSODAR (USCD1228), LSODPK (USCD1231), LSODKR (USCD1230), LSODI (USCD1224), LSOIBT (USCD1226), and LSODIS (USCD1225) which are distributed by the Computer Program Service of the NEA Data Bank.
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9. STATUS
Package ID Status date Status
USCD1226/01 23-SEP-2005 Tested at NEADB
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10. REFERENCES

[1]  A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE Solvers," in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64.
[2]  P. N. Brown and A. C. Hindmarsh, "Reduced Storage Matrix Methods in Stiff ODE Systems," J. Appl. Math. & Comp., 31 (1989), pp.40-91. 11.
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11. HARDWARE REQUIREMENTS
USCD1226/01
Compiling, loading, and executing the demonstration program required a minimum main storage of 6 Mbytes.
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12. PROGRAMMING LANGUAGE(S) USED
Package ID Computer language
USCD1226/01 FORTRAN-77
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13. SOFTWARE REQUIREMENTS
USCD1226/01
DOS under Windows XP.
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14. OTHER PROGRAMMING OR OPERATING INFORMATION OR RESTRICTIONS
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15. NAME AND ESTABLISHMENT OF AUTHORS

A.C. Hindmarsh* and L. R. Petzold
Lawrence Livermore National Laboratory
P.O. Box 808
Livermore, California 94550, USA
* Contact
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16. MATERIAL AVAILABLE
USCD1226/01
LSOIBT.inf  this info file
Double precision files:
DLSOIBT_MAIN.exe Executable file
DLSOIBT_MAIN.f Test Source file
DLSOIBT_OUT.aut Authors output file
DLSOIBT_OUT.nea NEA output file
opkda1.f  Fortran source file
opkda2.f  Fortran source file
opkdmain.f  Fortran source file
Single precision files:
opksa1.f  Fortran source file
opksa2.f  Fortran source file
opksmain.f  Fortran source file
SLSOIBT_MAIN.exe Executable file
SLSOIBT_MAIN.f Test source file
SLSOIBT_OUT.nea NEA output file
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17. CATEGORIES
  • P. General Mathematical and Computing System Routines

Keywords: algorithms, initial-value problems, numerical solution, ordinary differential equations.