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Partitioned second order method for magnetohydrodynamics in Elsässer variables
Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs
1. | Technical University of Munich, Faculty of Mathematics, 85748 Garching b. München, Germany |
2. | Département de Mathématiques et de Statistique, Université Laval, 1045 avenue de la Médecine, Québec, QC, G1V 0A6, Canada |
3. | McGill University, Department of Mathematics and Statistics, 805 Sherbrooke St W, Montreal, QC, H3A 0B9, Canada |
In this work, we introduce a method based on piecewise polynomial interpolation to enclose rigorously solutions of nonlinear ODEs. Using a technique which we call a priori bootstrap, we transform the problem of solving the ODE into one of looking for a fixed point of a high order smoothing Picard-like operator. We then develop a rigorous computational method based on a Newton-Kantorovich type argument (the radii polynomial approach) to prove existence of a fixed point of the Picard-like operator. We present all necessary estimates in full generality and for any nonlinearities. With our approach, we study two systems of nonlinear equations: the Lorenz system and the ABC flow. For the Lorenz system, we solve Cauchy problems and prove existence of periodic and connecting orbits at the classical parameters, and for ABC flows, we prove existence of ballistic spiral orbits.
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2 | 1556 | 14004 | 0.64 | |
3 | 1167 | 14004 | 0.58 |
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1 | 2333 | 13998 | ||
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1 | 2333 | 13998 | 0.97 | |
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1 | 2333 | 13998 | 0.97 | |
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19 | 233 | 13980 | 6.9 |
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