RKC swMATH ID: 808 Software Authors: Sommeijer, B.P.; Shampine, L.F.; Verwer, J.G. Description: RKC: An explicit solver for parabolic PDEs. An explicit Runge-Kutta-Chebychev algorithm for parabolic partial differential equations is discussed, implemented and tested. This method exploits some remarkable properties of a class of Runge-Kutta formulas of Chebychev type, proposed almost 20 year ago by P. J. van der Houwen and B. P. Sommeijer [Z. Angew. Math. Mech. 60, 479-485 (1980; Zbl 0455.65052)]. An s-stage (s≥2) method is discussed and analytical expressions for its coefficients are derived. An interesting property of this family makes it possible for the algorithm to select at each step the most efficient stable formula and the most efficient time-step. Various computational results and comparisons with other methods are provided. Homepage: http://www.netlib.org/ode/rkc.f Keywords: parabolic problems; time integration; stability; numerical examples; grid generation; reaction-diffusion equation; Runge-Kutta-Chebychev algorithm; Runge-Kutta formulas of Chebychev type Related Software: RODAS; VODE; S-ROCK; IRKC; SERK2; Matlab; SERK2v3; CVODE; MEBDF; PETSc; M3RK; PIROCK; DASSL; MATLAB expm; LSODA; ROWMAP; CHEMKIN; FEniCS; DRFM; LSODE Cited in: 132 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year RKC: An explicit solver for parabolic PDEs. Zbl 0910.65067Sommeijer, B. P.; Shampine, L. F.; Verwer, J. G. 1998 all top 5 Cited by 229 Authors 8 Verwer, Jan G. 7 González-Pinto, Severiano 7 Martín-Vaquero, Jesús 7 Najm, Habib N. 7 Sommeijer, Ben P. 6 Abdulle, Assyr 6 Pérez Rodríguez, S. 6 Zhang, Yongtao 5 Ray, Jaideep 4 Hernandez-Abreu, Domingo 4 Kleefeld, B. 4 Vilmart, Gilles 4 Weiner, Rüdiger 3 Almuslimani, Ibrahim 3 Bermejo, Rodolfo 3 Bochev, M. A. 3 Calhoun, Donna A. 3 Carpio, Jaime 3 Kleefeld, Andreas 3 Kleijn, Chris R. 3 Lefantzi, Sophia 3 Li, Jing-Rebecca 3 Safta, Cosmin 3 Seaïd, Mohammed 3 van Veldhuizen, S. 3 Vuik, Cornelis 2 Allan, Benjamin A. 2 Armstrong, Robert C. 2 Aslam, Tariq D. 2 Blaise, Sébastien 2 Cirilli, Stéphane 2 El-Amrani, Mofdi 2 Gear, Charles William 2 Guo, Daniel X. 2 Helzel, Christiane 2 Jiang, Tian 2 Khaliq, Abdul Q. M. 2 Knio, Omar M. 2 Li, Shoufu 2 Marzouk, Youssef M. 2 Medovikov, Alexei A. 2 Meyer, Chad D. 2 Montijano, Juan Ignacio 2 Niemeyer, Kyle E. 2 O’Sullivan, Stephen 2 Petzold, Linda Ruth 2 Prieto, Juan Luis 2 Sandu, Adrian 2 Shampine, Lawrence Fred 2 Sung, Chih-Jen 2 Wade, Bruce A. 2 Zhang, Hong 2 Zheng, Zheming 1 Ait-Haddou, Rachid 1 Alavani, Chitra 1 Alber, Mark S. 1 Alberti, Andrea 1 Alzahrani, H. H. 1 Asante-Asamani, E. O. 1 Balsara, Dinshaw S. 1 Beck, Steffen 1 Benson, Steven J. 1 Bergues Cabrales, Luis Enrique 1 Bernholdt, David E. 1 Bhatt, Harish P. 1 Bocher, Philippe 1 Bolaños Vaillant, Sorangel 1 Bruder, Jürgen 1 Brush, Lucien N. 1 Cai, Zhenning 1 Caliari, Marco 1 Castañeda, Antonio Rafael Selva 1 Charron, Martin 1 Chen, Shanqin 1 Cheng, Yuanzhen 1 Croci, Matteo 1 Cruz-Villar, Carlos Alberto 1 Dahlgren, Tamara L. 1 Das, Pratik 1 Dawes, Alan S. 1 De Souza, Giacomo Rosilho 1 Debusschere, Bert J. 1 Deeb, Ahmad 1 del Pozo, Josue Mariño 1 Del Sastre, P. Galán 1 Descombes, Stéphane 1 Diachin, Lori Freitag 1 Dormand, John R. 1 Duarte, Max 1 Dumont, Thierry 1 Eftekhari, Armin 1 Fahrenthold, Eric P. 1 Feodoritova, O. B. 1 Forbes, Lawrence K. 1 Foulon, S. 1 Frenklach, Michael 1 Freund, Jonathan B. 1 Galán del Sastre, Pedro 1 Ghazaryan, Anna 1 Glowinski, Roland ...and 129 more Authors all top 5 Cited in 39 Serials 34 Journal of Computational Physics 15 Journal of Computational and Applied Mathematics 8 SIAM Journal on Scientific Computing 7 BIT 7 Applied Numerical Mathematics 6 Applied Mathematics and Computation 6 Journal of Scientific Computing 3 Mathematics and Computers in Simulation 3 Computational Mathematics and Mathematical Physics 2 Computers & Mathematics with Applications 2 Computer Methods in Applied Mechanics and Engineering 2 Numerische Mathematik 2 Numerical Methods for Partial Differential Equations 2 Combustion Theory and Modelling 1 Computers and Fluids 1 Computer Physics Communications 1 Mathematics of Computation 1 International Journal for Numerical Methods in Engineering 1 SIAM Journal on Numerical Analysis 1 Mathematical and Computer Modelling 1 Applied Mathematical Modelling 1 SIAM Review 1 Communications in Numerical Methods in Engineering 1 Electronic Journal of Differential Equations (EJDE) 1 European Series in Applied and Industrial Mathematics (ESAIM): Proceedings 1 Discrete Dynamics in Nature and Society 1 International Journal of Theoretical and Applied Finance 1 The ANZIAM Journal 1 Computational Methods in Applied Mathematics 1 Comptes Rendus. 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Académie des Sciences, Paris 1 SIAM Journal on Applied Dynamical Systems 1 International Journal of Numerical Analysis and Modeling 1 Progress in Computational Fluid Dynamics 1 Acta Mechanica Sinica 1 Advances in Applied Mathematics and Mechanics 1 Journal of Numerical Mathematics and Stochastics 1 JSIAM Letters 1 Mathematics 1 SIAM/ASA Journal on Uncertainty Quantification all top 5 Cited in 24 Fields 115 Numerical analysis (65-XX) 46 Partial differential equations (35-XX) 29 Fluid mechanics (76-XX) 15 Ordinary differential equations (34-XX) 15 Classical thermodynamics, heat transfer (80-XX) 10 Biology and other natural sciences (92-XX) 7 Probability theory and stochastic processes (60-XX) 4 Computer science (68-XX) 4 Mechanics of deformable solids (74-XX) 3 Dynamical systems and ergodic theory (37-XX) 3 Optics, electromagnetic theory (78-XX) 3 Statistical mechanics, structure of matter (82-XX) 2 Integral equations (45-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Approximations and expansions (41-XX) 1 Functional analysis (46-XX) 1 Statistics (62-XX) 1 Mechanics of particles and systems (70-XX) 1 Geophysics (86-XX) 1 Operations research, mathematical programming (90-XX) 1 Information and communication theory, circuits (94-XX) Citations by Year