USCD1226 LSOIBT.
LSOIBT, Implicit Ordinary Differential Equations System Block Tridiagonal Matrices
NAME OR DESIGNATION OF PROGRAM, COMPUTER, DESCRIPTION OF PROGRAM OR FUNCTION, METHODS, RESTRICTIONS, CPU, FEATURES, RELATED OR AUXILIARY PROGRAMS, STATUS, REFERENCES, REQUIREMENTS, LANGUAGE, OPERATING SYSTEM, OTHER RESTRICTIONS, NAME AND ESTABLISHMENT OF AUTHORS, MATERIAL, CATEGORIES
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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.
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.
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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USCD1226/01
At the NEA-DB the demonstration program included in this package ran on a PC Windows Xeon in a few seconds.[ top ]
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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.
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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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.
[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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USCD1226/01
Compiling, loading, and executing the demonstration program required a minimum main storage of 6 Mbytes.[ top ]
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USCD1226/01
LSOIBT.inf this info fileDouble 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
Keywords: algorithms, initial-value problems, numerical solution, ordinary differential equations.