A Comparison of Several Formulas on Lightly Damped, Oscillatory Problems

Second derivative methods applied to implicit first‐ and second‐order systems

“…Turning now to the question of stability, it is conventional to study the application of the method to the homogeneous constant coefficient problem, with A real, y' = Ay (7) where A is assumed diagonalizable and where the eigenvalues A, of A satisfy Real (A,) 4 0. It is sufficient then to consider the scalar equation y' = Ay, with Real ( A ) S 0 and integration in a positive direction, and for this equation to reqiire that the numerical method (2) simulate the decaying behaviour of the solution by requiring that Iypl G /ys-ll for all s. If this is true for all Ah such that Real (Ah) S 0, we say that the formula is A-stable; this is a property of all the five second derivative methods considered in the next section.…”

“…, n and the general solution of (4) contains terms eAir which can only be satisfactorily simulated using an L-stable method (or at least a method with the corresponding property for real Ah, namely &-stability, a property whose definition is clear from Lambert.6) The problem ( 5 ) is rather more complicated. If we assume, as is usually the case, that M, C and K are all symmetric and non-negative definite, with M positive definite, then we are guaranteed'3x14 that the eigenvalues of A (resulting from writing ( 5 ) as a first-order system (7)) comprise non-positive real eigenvalues and complex conjugate pairs with non-positive real parts. We observe that if C = 0 the eigenvalues are all pure imaginary, if C is small (weak damping) the eigenvalues occur in complex conjugate pairs and if C is large (strong damping) the eigenvalues are all real.…”

An Investigation into the Stability Properties of Second Derivative Methods Using Perfect Square Iteration Matrices

“…For example, Enright [6] factors W into -To( rI hJ)( I hJ) where r -(flo/2 7o) + i( (flo/2 To)2 (1/ To) 1/2, and uses complex arithmetic. This modification has the disadvantage of requiring twice as much storage and being up to four times slower than real arithmetic (see Addison [2]). An alternative modification, suggested by Skeel and Kong [12] in the context of blended linear multistep methods, is the approximation of the iteration matrix W by a perfect square matrix (I-chJ)2, where c is real and is chosen to minimize the rate of convergence of the iteration scheme applied to the test equation A third modification, and the one considered here, is the approximation of W by a perfect square matrix (aI-bhJ)2, where a and b are chosen to yield iteration schemes with the best possible absolute stability properties.…”

“…This is important because it suggests that instability could become a major problem (in the best of situations) if several successive steps of the integration are taken with only one iteration per step. Two iterations of the perfect square approach require 4n 2 real operations plus 4 function evaluations, whereas one iteration of the complex arithmetic approach requires the equivalent of at most 4n 2 real operations plus 2 function evaluations (see Addison [2]). Consequently, if a minimum of two iterations is required with the perfect square approach for stability reasons, the cost of performing the iterations, independent of matrix factorizations, may be more expensive than using complex arithmetic.…”