ISSN:
0945-3245
Keywords:
AMS(MOS): 65L07
;
CR: 5.17
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary Consider the systemy′=f(x,y),y(a)=η,x∈[a,b],y∈R n wheref is continuous and Lipschitzian with respect to the second argument. Very often linear multistep variable stepsize variable formula methods (LM VSVFM's) are used to computey k≈y(xk) on the points of the grid:a=x 0〈x1〈x2〈...〈xN=b. The general LM VSVFM is based on formulae of the following type $$y_k = \sum\limits_{i = 1}^{s_k } {\alpha _i (\bar h_k ,sk)y_{k - i} } + \sum\limits_{i = 0}^{s_k } {h_{k - i} \beta _i (\bar h_k ,s_k )f(x_{k - i} ,y_{k - i} )} $$ whereh k=xk−xk−1, $$\bar h_k = (h_k ,h_{k - 1} , \ldots ,h_{k - s_k } )$$ ,s k≦k, k=1(1)N. The coefficients α i and β i depend on the lasts k+1 stepsizes and on the formula used at stepk. Only the zero-stability properties of some special classes of LM VSVFM's (as for example those based on Adams formulae) were investigated in the literature. A class of three-ordinate LM VSVFM's is defined in this paper. Some results concerning the zero-stability properties of these methods are proved. It is shown that some well-known results are simple corollaries of the results found for the three-ordinate LM VSVFM's. It is easily seen that similar results hold for the corresponding one-leg VSVFM's. Finally, the use of the theoretical results in the practical implementations is briefly discussed.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01398250
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