High order explicit methods for parabolic equations

S. N. Bernstein,Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d'une variable réelle, in Professées á la Sorbonne, Paris, 1926.

E. Hairer, S. P. Nørsett, and G. Wanner,Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd ed., Springer-Verlag, Berlin, 1993.

E. Hairer and G. Wanner,Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd ed., Springer-Verlag, Berlin, 1996.

P. J. van der Houwen and J. Kok,Numerical solution of a maximal problem, Report TW 124/71, Mathematical Center, Amsterdam, 1971.

P. J. van der Houwen,Construction of Integration Formulas for Initial Value Problems, North-Holland, Amsterdam, 1977.

P. J. van der Houwen and B. P. Sommeijer,On the internal stability of explicit m-stage Runge-Kutta methods for large m-values, Z. Angew. Math. Mech., 60 (1980), pp. 479–485.

V. I. Lebedev and A. A. Medovikov,Methods of second order accuracy with variable time steps, Izvestiya Vuzov, Matematika, N10, Russia, 1995.

V. I. Lebedev,How to solve stiff systems of differential equations by explicit methods, in Numerical Methods and Applications, CRC Press, Boca Raton, 1994, pp. 45–80.

V. I. Lebedev,Zolotarev polynomials and extremum problem, Russ. J. Numer. Anal. Math. Modelling. V. 9, N 3 (1994), pp. 191–314.

V. I. Lebedev,A new method for determing the roots of polynomials of least deviation on a segment with weight and subject to additional conditions. Part I, Part II, Russ. J. Numer. Anal. Math. Modelling. V. 8, N 3 (1993), pp. 195–222; V. 8, N 5 (1993), pp. 397–426.

V. I. Lebedev,Explicit difference schemes with time-variable steps for the solution of stiff system of equations, Preprint DNM AS USSR N177, 1987.

H. Lomax,On the construction of highly stable, explicit numerical methods for integrating coupled ordinary differential equations with parasitic eigenvalues, NASA Technical Note NASATND/4547, 1968.

A. A. Markov,Lectures on the functions of least deviation, 1892; in Selected Works, OGIZ, Moscow, 1948, pp. 244–291, (in Russian).

A. A. Medovikov,Explicit and explicit-implicit methods of solution of unstationary problems, PhD Thesis, INM, Moscow, 1992.

C. L. Metzger,Méthodes de Runge-Kutta de rang supérieur à l'orde, Thèse (troisième cycle), Université de Grenoble, France, 1967.

W. Riha,Optimal stability polynomials, Computing, 9 (1972), pp. 37–43.

G. Szegö,Orthogonal Polynomials, Amer. Math. Soc., New York, 1949.

P. L. Chebyshev,On the problems about minimal values connected with appropriate representation of functions, Mem. Acad. St. Petersburg (1859);

Collected Works, Vol. 2, GITTL, Moscow-Leningrad, 1947 (in Russian); Oeuvres, Vol. I, p. 271.

P. L. Chebyshev,On the functions slightly deviating from zero for certain values of the variables, Prilojenie k XI tomy Zapisok Imperatorskoi Akademii nauk N3 (1881);

Collected Works, Vol. 3, GITTL, Moscow-Leningrad, 1948, p. 108; Oeuvres, Vol. II, p. 373 (in Russian).

J. G. Verwer,A class of stabilized three-step Runge-Kutta methods for the numerical integration of parabolic equations, J. Comput. Appl. Math., 3 (1977), pp. 155–166.

J. G. Verwer,Explicit Runge-Kutta methods for parabolic partial differential equations, 1996. To appear in the Runge-Kutta centennial special issue of Appl. Numer. Math.

E. I. Zolotarev,On applying elliptic functions to the study of functions of least and greatest deviation from zero, Zapiski Sankt-Peterburgskoi Akademii Nauk XXX, No 5 (1887), pp. 1–59 (in Russian).