Adapted BDF Algorithms: Higher-order Methods and Their Stability

We present BDF type formulas of high-order (4, 5 and 6), capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential with parameter A and ordinary polynomials. For A = 0, the new formulas reduce to the classical BDF formulas. Theorems of the local truncation error reveal the good behavior of the new methods with stiff problems. Plots of their 0-stability regions in terms of the eigenvalues of the parameter Ah are provided. Plots of their absolute stability regions that include the whole of the negative real axis are provided. The weights of the method usually require the evaluation of a matrix exponential. However, if the dimension of the matrix is large, we shall not perform this calculus and shall only approximate those coefficients once. Numerical examples underscore the efficiency of the proposed codes, especially when one is integrating stiff oscillatory problems.