## Stable implicit and explicit numerical methods for integrating quasi-linear differential equations with parasitic-stiff and parasitic-saddle eigenvalues |

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accuracy and stability accurate Adams-Moulton Ames Research Center An]w Asymptotic autonomous calculation step characteristic equation complex plane constant coefficients constants dependent coupled ordinary differential Dahlquist's theorem data at previous defined by equations difference equation equa equation 53 equations with constant equations with parasitic EQUATIONS WITH PARASITIC-STIFF equispaced error example Germund G given by equation given in reference h[An HIGHLY STABLE EXPLICIT implicit methods initial conditions Integrating Coupled Locus of computed Math method defined METHODS FOR INTEGRATING modified Euler method Moffett Field Multistep NASA TN numerical integration operation order of accuracy order Runge-Kutta method Ordinary Differential Equations parasitic eigenvalues plane QUASI-LINEAR DIFFERENTIAL EQUATIONS Real stability boundary saddle curve saddle point set of coupled set of differential shown in sketch single root Sketch f Sketch n spurious roots stable explicit methods step number Taylor series expansion tions Treanor unstable wn+1 zero