In this paper, we consider the construction of explicit General Linear Methods (GLM) for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The discrete coefficients of the methods are obtained from the continuous coefficients of the continuous schemes which are derived using collocation and interpolation approach. One of the new method's stability matrix M(z) has a single nonzero eigenvalue similar to that of the third order Runge Kutta method (RKM), while other eigenvalues being zero, such general linear method is referred to as GLM with Runge Kutta stability. Numerical results are provided to illustrate the application of the new schemes. The new schemes are found to have competitive accuracy with the existing Almost Runge Kutta Methods (ARKM) discussed in Butcher [9].
ButcherJ. C., A modified multistep methods for the numerical integration of ordinary differential equations. J. ACM12, No. 1 (1965), 125–135.
2.
ButcherJ. C., A multistep generalization of Runge Kutta methods with four or five stages. J. ACM14, No. 1 (1967), 84–99.
3.
ButcherJ. C., An algebraic theory of integration methods for ordinary differential equations. Maths. Comp., 26, (1972), 79–106.
4.
BurrageK. and ButcherJ.C., Non-linear stability of a general class of differential equation methods, BIT20, (1980), 185–203.
5.
ButcherJ. C., General linear methods: Survey., Applied Numer. Maths.1, (1985), 273–284.
6.
ButcherJ. C., The Numerical Analysis of Ordinary Differential Equation: Runge Kutta and General Linear Methods, Wiley, Chichester, 1987.
7.
ButcherJ. C., Some new hybrid methods for IVPs. In. CashJ. R. and WellGlad (eds), Computational ODEs.Clarendon press, Oxford (1992), 29–46.
8.
ButcherJ. C., General linear methods. Comp. Maths. Applic.31, No. 4/5 (1996), 106–112.
9.
ButcherJ. C., ARK methods up to order five. Numerical Algorithms17 (1998), pp. 193–221.
10.
ButcherJ. C., Numerical Methods for Ordinary Differential Equations.Second Edition., J. Wiley, Chichester, (2008).
11.
ButcherJ. C., General linear methods for ordinary differential equations. Maths and Comp. in Simulation, 79 (2009), 1834–1845.
12.
ButcherJ. C., Trees and numerical methods for ordinary differential equations. Numerical Algorithms, 53 (2010), 153–170.
13.
DahlquistG., A special stability problem for linear multistep methods, BIT, 3, (1963), pp. 27–43.
14.
EnrightW. H., Continuous numerical methods for ODEs with defect control. J. Comput. Appl. Math., 125 (2000), pp. 159–170.
15.
GraggW. B. and StetterH.J., Generalized multistep predictor-corrector methods. J. Assoc. Mach., 11, (1964), 188–209.
16.
GearC. W., Hybrid methods for initial value problems in ordinary differential equations. SIAM. J. Numer. Anal., 2, (1965), 69–86.
17.
HairerE.NorsetS.P., and WannerG., Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, (1993).
18.
HairerE., and WannerG., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, (1996).
19.
HighamJ. D. and HighamJ.N., Matlab Guide.SIAM. Philadelphia (2000).
20.
IkhileM. N. O., and OkuonghaeR. I., Stiffly stable continuous extension of second derivative LMM with an off-step point for IVPs in ODEs. J. Nig. Assoc. Math. Physics.Vol. 11 (2007), pp. 175–190.
21.
JackiewiczZ.VermiglioR., and ZennaroM., Variable stepsize diagonally implicit multistage integration methods for ODEs. Comp. and Maths with Appli.47 (2004) 1463–1475.
22.
LieI., and NorsetS.P., Superconvergence for multistep collocation Maths, of Comp.52, No. 185, (1989), 65–79.
23.
OkuonghaeR. I., Stiffly Stable Second Derivative Continuous LMM for IVPs in ODEs. Ph.D Thesis. Dept. of Maths. University of Benin, Benin City. Nigeria. (2008).
24.
OkuonghaeR. I., A class of continuous hybrid LMM for stiff IVPs in ODEs. Analele Stiintiifice ale Universitat II AL.I. Cuza Din lasi (S. N.)MATEMATICA, Tomul LVIII, (2012), f.2, pp.239–258.
25.
OkuonghaeR. I. and IkhileM.N.O., A continuous formulation of A(α)-stable second derivative linear multistep methods for stiff IVPs and ODEs. J. of Algorithms and Comp. Technology, Vol. 6, No. 1 (2011), 79–101.
26.
SelvaM.ArevaloC., and FuhererC., A collocation formulation of multistep methods for variable step-size extensions. Appl. Numer. Math, Vol. 42 (2002). pp. 5–16.
