The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
High order algebraically stable multistep Runge-Kutta methods
SIAM Journal on Numerical Analysis
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
A general class of two-step Runge-Kutta methods for ordinary differential equations
SIAM Journal on Numerical Analysis
Derivation of continuous explicit two-step Runge-Kutta methods of order three
Journal of Computational and Applied Mathematics
Multistep collocation methods for Volterra Integral Equations
Applied Numerical Mathematics
Algebraically stable diagonally implicit general linear methods
Applied Numerical Mathematics
Two-step Runge-Kutta Methods with Quadratic Stability Functions
Journal of Scientific Computing
Mathematics and Computers in Simulation
Two-step diagonally-implicit collocation based methods for Volterra Integral Equations
Applied Numerical Mathematics
Two-step modified collocation methods with structured coefficient matrices
Applied Numerical Mathematics
Search for highly stable two-step Runge-Kutta methods
Applied Numerical Mathematics
Exponentially fitted singly diagonally implicit Runge-Kutta methods
Journal of Computational and Applied Mathematics
P-stable general Nyström methods for y˝=f(y(t))
Journal of Computational and Applied Mathematics
Order conditions for General Linear Nyström methods
Numerical Algorithms
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We investigate algebraic stability of the new class of two-step almost collocation methods for ordinary differential equations. These continuous methods are obtained by relaxing some of the interpolation and collocation conditions to achieve strong stability properties together with uniform order of convergence on the whole interval of integration. We describe the search for algebraically stable methods using the criterion based on the Nyquist stability function proposed recently by Hill. This criterion leads to a minimization problem in one variable which is solved using the subroutine fminsearch from MATLAB. Examples of algebraically stable methods in this class are also presented.