On the use of stability regions in the numerical analysis of initial value problems

This paper deals with the stability analysis of one-step methods in the numerical solution of initial (-boundary) value problems for linear, ordinary, and partial differential equations. Restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size. These restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration. The errors in the one-step methods are measured in arbitrary norms (not necessarily generated by an inner product). The theory is illustrated in the numerical solution of the heat equation and some other differential equations, where the error growth is measured in the maximum norm.

[1]  F. Smithies.  Linear Operators , 2019, Nature.

[2]  Seymour V. Parter,et al.  Stability, convergence, and pseudo-stability of finite-difference equations for an over-determined problem , 1962 .

[3]  H. Schneider,et al.  The bauer fields of values of a matrix , 1964 .

[4]  J. Duncan,et al.  Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras: Hermitian elements of a complex unital Banach algebra , 1971 .

[5]  F. Bonsall,et al.  Numerical Ranges II , 1973 .

[6]  I. M. Glazman,et al.  Finite-Dimensional Linear Analysis: A Systematic Presentation in Problem Form , 1974 .

[7]  Olavi Nevanlinna,et al.  On the numerical integration of nonlinear initial value problems by linear multistep methods , 1977 .

[8]  Germund Dahlquist,et al.  G-stability is equivalent toA-stability , 1978 .

[9]  V. Thomée,et al.  ON RATIONAL APPROXIMATIONS OF SEMIGROUPS , 1979 .

[10]  Marie-Noëlle Le Roux,et al.  Semidiscretization in time for parabolic problems , 1979 .

[11]  K. W. Morton.  Stability of finite difference approximations to a diffusion–convection equation , 1980 .

[12]  David F. Griffiths,et al.  Analysis of error growth for explicit difference schemes in conduction–convection problems , 1980 .

[13]  Remarks on Time Discretization of Contraction Semigroups , 1985 .

[14]  M. N. Spijker.  Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems , 1985 .

[15]  Gustaf Söderlind,et al.  Bounds on nonlinear operators in finite-dimensional banach spaces , 1986 .

[16]  Gustaf Stiderlind.  Bounds on nonlinear operators in finite-dimensional banach spaces , 1986 .

[17]  M. Crouzeix.  On multistep approximation of semigroups in Banach spaces , 1987 .

[18]  M. N. Spijker,et al.  Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations , 1987 .

[19]  Lloyd N. Trefethen,et al.  Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues , 1990 .

[20]  M. N. Spijker,et al.  A generalization of the numerical range of a matrix , 1990 .

[21]  M. N. Spijker,et al.  On a generalization of the resolvent condition in the Kreiss matrix theorem , 1991 .