Algebraic criteria for A-stability of peer two-step methods
For numerical analysts designing stable time integration methods, this work offers exact algebraic criteria and constructive parametrization, though it is incremental over known stability theory for General Linear Methods.
The paper presents algebraic criteria for A-stability of peer two-step methods, providing an explicit parametrization of A-stable methods of order s-1 with a weight matrix as parameter. It constructs diagonally-implicit and parallel L-stable peer methods with 3 and 4 stages.
A new criterion for A-stability of peer two-step methods is presented which is verifiable exactly in exact arithmetic by checking semi-definiteness of a certain test matrix. It depends on the existence of two positive definite weight matrices for a given method. Although the initial approach is different using properties of the numerical radius the criterion itself resembles the one from algebraic stability of General Linear Methods. Known numerical algorithms for the computation of the unknown weight matrices suffer from rank deficiencies of the test matrix. For $s$-stage peer methods of order $s-1$ this rank defect is identified with an explicit block diagonal decomposition of the test matrix in trivial and definite blocks. In the design of methods its coefficients are unknown and an explicit parametrization of A-stable peer methods of order $s-1$ is presented with a weight matrix as parameter. This leads to a general existence result for any number of stages. The restrictions for efficient L-stable peer methods like diagonally-implicit and parallel ones are also discussed and such methods with $3$ and $4$ stages are constructed.