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Analytical Jacobian-vector products for the matrix-free time integration of partial differential equations. (English) Zbl 1348.65109
Summary: Many scientific and engineering applications require the solution of large systems of initial value problems arising from method of lines discretization of partial differential equations. For systems with widely varying time scales, or with complex physical dynamics, implicit time integration schemes are preferred due to their superior stability properties. These schemes solve at each step linear systems with matrices formed using the Jacobian of the right hand side function. For large applications iterative linear algebra methods, which make use of Jacobian-vector products, are employed. This paper studies the impact that the method of computing Jacobian-vector products has on the overall performance and accuracy of the time integration process. The analysis shows that the most beneficial approach is the direct computation of exact Jacobian-vector products in the context of matrix-free time integrators. This approach does not suffer from approximation errors, reuses the parallelism and data distribution already present in the right-hand side vector computations, and avoids storing or operating on the entire Jacobian matrix.

65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Hairer, E.; Wanner, G., Solving ordinary differential equations II: stiff and differential-algebraic problems, (2002), Springer
[2] Knoll, D.; Keyes, D., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 2, 357-397, (2004), URL http://www.sciencedirect.com/science/article/pii/S0021999103004340 · Zbl 1036.65045
[3] Lang, J.; Verwer, J., Ros3p—an accurate third-order rosenbrock solver designed for parabolic problems, BIT, 41, 4, 731-738, (2001) · Zbl 0996.65099
[4] Liao, W., A strongly a-stable time integration method for solving the nonlinear reaction-diffusion equation, (Abstract and Applied Analysis, Vol. 2015, (2015), Hindawi Publishing Corporation) · Zbl 1352.65204
[5] Weiner, R.; Schmitt, B.; Podhaisky, H., ROWMAP — a ROW-code with Krylov techniques for large stiff odes, Appl. Numer. Math., 25, 2-3, 303-319, (1997), URL http://www.sciencedirect.com/science/article/pii/S0168927497000676. Special Issue on Time Integration · Zbl 0895.65035
[6] Tranquilli, P.; Sandu, A., Rosenbrock-Krylov methods for large systems of differential equations, SIAM J. Sci. Comput., 36, 3, A1313-A1338, (2014) · Zbl 1320.65108
[7] Hascoët, L.; Pascual, V., The tapenade automatic differentiation tool: principles, model, and specification, ACM Trans. Math. Software, 39, 3, 20:1-20:43, (2013) · Zbl 1295.65026
[8] Giering, R.; Kaminski, T., Recipes for adjoint code construction, ACM Trans. Math. Softw., 24, 4, 437-474, (1998) · Zbl 0934.65027
[9] Bischof, C. H.; Carle, A.; Khademi, P.; Mauer, A., ADIFOR 2.0: automatic differentiation of Fortran 77 programs, IEEE Comput. Sci. Eng., 3, 3, 18-32, (1996)
[10] Liska, R.; Wendroff, B., Composite schemes for conservation laws, SIAM J. Numer. Anal., 35, 6, 2250-2271, (1998) · Zbl 0920.65054
[11] OpenMP Architecture Review Board, OpenMP application program interface version 4.0, 2013. URL http://www.openmp.org/mp-documents/OpenMP4.0.0.pdf.
[12] Tranquilli, P.; Glandon, R.; Sandu, A., CUDA acceleration of a matrix-free rosenbrock-K method applied to the shallow water equations, (Proceedings of the Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems, ScalA ’13, (2013), ACM New York, NY, USA), 5:1-5:6, URL http://doi.acm.org/10.1145/2530268.2530273
[13] Allen, S. M.; Cahn, J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27, 6, 1085-1095, (1979), URL http://www.sciencedirect.com/science/article/pii/0001616079901962
[14] Rang, J.; Angermann, L., New rosenbrock methods of order 3 for PDAEs of index 2, Adv. Differ. Equ. Control Process., 1, 2, 193-217, (2008) · Zbl 1162.65386
[15] Tranquilli, P.; Sandu, A., Exponential-Krylov methods for ordinary differential equations, J. Comput. Phys., 278, 31-46, (2014), URL http://www.sciencedirect.com/science/article/pii/S0021999114005592 · Zbl 1349.65228
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