Stability and Multiplicity of Solutions to Discretizations of Nonlinear Ordinary Differential Equations

A large class of consistent and unconditionally stable discretizations of nonlinear boundary value problems is defined. The number of solutions to the discretizations is compared to the number of solutions of the continuous problem. We state conditions under which these numbers must agree for all sufficiently small mesh sizes. Various examples, including bifurcation problems, illustrate our theoretical results.

[1]  E. Doedel.  Some stability theorems for finite difference collocation methods on nonuniform meshes , 1980 .

[2]  Thomas R. Lucas,et al.  Some Collocation Methods for Nonlinear Boundary Value Problems , 1972 .

[3]  Heinz-Otto Kreiss,et al.  Difference approximations for boundary and eigenvalue problems for ordinary differential equations , 1972 .

[4]  M. R. Osborne,et al.  A method for finite-difference approximation to ordinary differential equations , 1964, Computer/law journal.

[5]  G. W. Reddien,et al.  Projection Methods for Two-Point Boundary Value Problems , 1980 .

[6]  Collocation, Difference Equations, and Stitched Function Representations , 1975 .

[7]  On the convergence of the finite difference method for nonlinear ordinary boundary value problems , 1979 .

[8]  John R. Rice,et al.  A high-order difference method for differential equations , 1980 .

[9]  U. Ascher,et al.  A collocation solver for mixed order systems of boundary value problems , 1979 .

[10]  Blair Swartz.  Compact, implicit difference schemes for a differential equation’s side conditions , 1980 .

[11]  L. Shampine,et al.  A collocation method for boundary value problems , 1972 .

[12]  H. Keller,et al.  Approximation methods for nonlinear problems with application to two-point boundary value problems , 1975 .

[13]  C. D. Boor,et al.  Collocation at Gaussian Points , 1973 .

[14]  On the Stability of Boundary Conditions for Separable Difference Approximations to Parabolic Equations , 1977 .

[15]  Eusebius J. Doedel,et al.  FINITE DIFFERENCE COLLOCATION METHODS FOR NONLINEAR TWO POINT BOUNDARY VALUE PROBLEMS , 1979 .

[16]  E. Bohl,et al.  On the bifurcation diagram of discrete analogues for ordinary bifurcation problems , 1979 .

[17]  R. Grigorieff,et al.  Diskrete Approximation von Eigenwertproblemen , 1975 .