On the implementation of singly implicit Runge-Kutta methods

Author:
G. J. Cooper

Journal:
Math. Comp. **57** (1991), 663-672

MSC:
Primary 65L06; Secondary 65Y05

DOI:
https://doi.org/10.1090/S0025-5718-1991-1094945-2

MathSciNet review:
1094945

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A modified Newton method is often used to solve the algebraic equations that arise in the application of implicit Runge-Kutta methods. When the Runge-Kutta method has a coefficient matrix *A* with a single point spectrum (with eigenvalue ), the efficiency of the modified Newton method is much improved by using a similarity transformation of *A*. Each iteration involves vector transformations. In this article an alternative iteration scheme is obtained which does not require vector transformations and which is simpler in other respects also. Both schemes converge in a finite number of iterations when applied to linear systems of differential equations, but the new scheme uses the nilpotency of to achieve this. Numerical results confirm the predicted convergence for nonlinear systems and indicate that the scheme may be a useful alternative to the modified Newton method for low-dimensional systems. The scheme seems to become less effective as the dimension increases. However, it has clear advantages for parallel computation, making it competitive for high-dimensional systems.

**[1]**T. A. Bickart,*An efficient solution process for implicit Runge-Kutta methods*, SIAM J. Numer. Anal.**14**(1977), 1022-1027. MR**0458893 (56:17092)****[2]**K. Burrage,*A special family of Runge-Kutta methods for solving stiff differential equations*, BIT**18**(1978), 22-41. MR**0483458 (58:3459)****[3]**-,*Efficiently implementable algebraically stable Runge-Kutta methods*, SIAM J. Numer. Anal.**19**(1982), 245-258. MR**650049 (83d:65235)****[4]**K. Burrage, J. C. Butcher, and F. H. Chipman,*An implementation of singly-implicit Runge-Kutta methods*, BIT**20**(1980), 326-340. MR**595213 (82h:65049)****[5]**J. C. Butcher,*On the implementation of implicit Runge-Kutta methods*, BIT**16**(1976), 237-240. MR**0488746 (58:8263)****[6]**-,*Some implementation schemes for implicit Runge-Kutta methods*(Proc. Dundee Conf. on Numerical Analysis 1979), Lecture Notes in Math., vol. 773, Springer-Verlag, pp. 12-24. MR**569458 (81g:65096)****[7]**J. R. Cash,*On a class of implicit Runge-Kutta procedures*, J. Inst. Math. Appl.**19**(1977), 455-470. MR**0436597 (55:9540)****[8]**F. H. Chipman,*The implementation of Runge-Kutta implicit processes*, BIT**13**(1973), 391-393. MR**0337009 (49:1782)****[9]**A. G. Collings and G. J. Tee,*An analysis of Euler and implicit Runge-Kutta numerical integration schemes for structural dynamic problems*, Proc. Sixth Australasian Conf. on the Mechanics of Structures and Materials 1977, vol. 1, pp. 147-154.**[10]**G. J. Cooper and J. C. Butcher,*An iteration scheme for implicit Runge-Kutta methods*, IMA J. Numer. Anal.**3**(1983), 127-140. MR**716457 (85b:65060)****[11]**H. T. Davis,*Introduction to nonlinear differential and integral equations*, Dover, New York, 1962. MR**0181773 (31:6000)****[12]**W. H. Enright,*Improving the efficiency of matrix operations in the numerical solution of ODEs*, Technical Report no. 98, Computer Science Dept., Univ. of Toronto, 1976.**[13]**R. Frank and C. W. Ueberhuber,*Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations*, BIT**17**(1977), 146-159. MR**0488748 (58:8265)****[14]**C. W. Gear,*The automatic integration of stiff ordinary differential equations*, Proc. IFIP Congress 1968, pp. 187-193. MR**0260180 (41:4808)****[15]**E. Hairer and G. Wanner,*Algebraically stable and implementable Runge-Kutta methods of high order*, SIAM J. Numer. Anal.**18**(1981), 1098-1108. MR**639000 (82k:65038)****[16]**S. P. Nørsett,*Runge-Kutta methods with a multiple real eigenvalue only*, BIT**16**(1976), 388-393. MR**0440928 (55:13796)****[17]**J. M. Varah,*On the efficient implementation of implicit Runge-Kutta methods*, Math. Comp.**33**(1979), 557-561. MR**521276 (80b:65105)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L06,
65Y05

Retrieve articles in all journals with MSC: 65L06, 65Y05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1991-1094945-2

Article copyright:
© Copyright 1991
American Mathematical Society