2017/01/03 by Andreas Griewank, Griewank, Andreas, Richard Hasenfelder +5
Computer Science · Mathematics · Physics and Astronomy · #65L05 #65L06 #65L70 #65L99 #65P10 #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1701.00745
openalex publication_date 2017/01/03 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
In this article we analyze a generalized trapezoidal rule for initial value\nproblems with piecewise smooth right hand side (F: Rn\→ Rn ). When applied\nto such a problem the classical trapezoidal rule suffers from a loss of\naccuracy if the solution trajectory intersects a nondifferentiability of (F ).\nThe advantage of the proposed generalized trapezoidal rule is threefold:\nFirstly we can achieve a higher convergence order than with the classical\nmethod. Moreover, the method is energy preserving for piecewise linear\nHamiltonian systems. Finally, in analogy to the classical case we derive a\nthird order interpolation polynomial for the numerical trajectory. In the\nsmooth case the generalized rule reduces to the classical one. Hence, it is a\nproper extension of the classical theory. An error estimator is given and\nnumerical results are presented.\n