V.I.Lebedev
Russian Research Center ``Kurchatov Institute'',
Institute of Numerical Mathematics,
Moscow, Russia,
nucrect@inm.ras.ru
Problem. In applications a necessity often arises to solve Cauchy problems for stiff differential equations or equations derived from the method of lines [1], when the use of implicit schemes is sophisticated, and the time step in explicit schemes is too short.
The core. To solve Cauchy problems for stiff differential equations or equations derived from the method of lines [2, 3, 4], explicit stable difference schemes with the steps varying in time were proposed. The stability conditions for the optimal algorithm of choosing stepsize were investigated. The algorithm provides a drastic improvement compared with known explicit schemes (up to 30000000 in the case of real spectrum). A special algorithm is used for interior and boundary layers. Algorithms are based on the properties of T-sequence of roots of Chebyshev polynomials [5,6,7].
The advantages of the proposed method. The methods are convenient to implement on parallel and pipeline computers, suitable to solve highly multidimensional problems, problems with nonlinear, nonsymmetric, indefinite operators, for unconstrained optimization. Adaptive properties of the method consist in the possibility of choice of:
The code. The method is implemented in FORTRAN code DUMKA. Computations were conducted on CONVEX, CYBER, ELBRUS, BESM and different PCs.
The method is quite flexible. The use in applications is simple, since DUMKA user should just write a subroutine for right-hand side of equations and a subroutine, estimating the greatest in modulus negative eigenvalue of Jacobian. The method requires the storage of only three arrays of variables.
Testing of the code. To test and analyze the code the standard testing set was used [10, 11]. The problems were divided into six groups by the sort of spectrum (real, complex), and the sort of equations (linear, nonlinear). 12 stiff test problems from [12] were solved by DUMKA. Time of solution and accuracy were compared to those of solvers: RKC, RADAU5, LSODE, EPISODE, DOPRI8(5), SDIRK4, SEVLEX, SODEX, ROS4, RODAS, STRIDE. The results of testing confirmed the suitability of using DUMKA for integration of stiff systems.
Additionally some problems, not included into the sets [10, 11, 12], but of interest in investigation of the ability of the code to solve neutron transport problems were solved, and also some problems from [4] (linear heat transfer equation for three-dimensional domain with a large number of nodes, a system of multigroup, one-dimensional kinetic equations with delayed neutrons, Babuska's example, elasticity equation and so on). These results confirmed the assumptions about the features of the method and advantages of the code in solution of stiff systems.
References.
1. Lebedev V.I. Equations and convergence of the differential difference method (the method of lines). Vestnik MGU, No.10, pp.47-57, 1995 (in Russian).
2. Lebedev V.I. Explicit difference schemes with time-variable steps for solution of stiff systems of equations. Preprint DNM AS USSR No.177, 1987.
3. Lebedev V.I. Explicit difference schemes with time-variable steps for solution of stiff systems of equations. Sov. J. Numer. Anal. Math. Modelling. Vol.4, No.2, pp.111-135, 1989.
4. Lebedev V.I. How to solve stiff systems of equations by explicit difference schemes. Numerical methods and applications, Ed. G.I.Marchuk, CRC Press, Boca Raton, Ann Arbor, London, Tokyo, pp.45-80, 1994.
5. Lebedev V.I., Finogenov S.A. On the utilization of ordered Chebyshev parameters in iterative methods. Zh. Vychisl.Mat.Mat.Phys, Vol.16, No.4, pp.895-907, 1976 (in Russian).
6. Lebedev V.I. An Introduction to Functional Analysis and Computational Mathematics. Birkhäuser, Boston, Basel, Berlin, 1996.
7. Lebedev V.I. Zolotarev polynomials and extremum problems. Russ. J. Numer. Anal. Math. Modelling, Vol.9, No.3, pp.231-263, 1994.
8. Medovikov A.A. Third order explicit method for the stiff ordinary differential equations. Lecture Notes in Computer Science 1196. Numerical Analysis and Its Applications. Springer. pp.327-334. 1997.
9. Lebedev V.I. Explicit difference schemes with variable time steps for solving stiff systems of equations. Lecture Notes in Computer Science 1196. Numerical Analysis and Its Applications. Springer. pp.274-283, 1997.
10. Enright W.H.,Pryce J.D. Two FORTRAN Packages for Assessing Initial Value Methods. ACM Trans. Math. Soft., Vol.13, No.1, pp.1-27, 1987.
11. Byrne G.D.,Hindmarsh A.C. Stiff ODE Solvers:A Review of Current and Coming Attractions. J. of Comp. Physics. Vol.70, pp.1-62(1987).
12.E. Hairer, G. Wanner. Solving Ordinary Differential Equations II. Second Revised Edition, Springer-Verlag, 1996.