SERK2v2: A new second-order stabilized explicit Runge-Kutta method for stiff problems. Traditionally, explicit numerical algorithms have not been used with stiff ordinary differential equations (ODEs) due to their stability. Implicit schemes are usually very expensive when used to solve systems of ODEs with very large dimension. Stabilized Runge-Kutta methods (also called Runge-Kutta-Chebyshev methods) were proposed to try to avoid these difficulties. The Runge-Kutta methods are explicit methods with extended stability domains, usually along the negative real axis. They can easily be applied to large problem classes with low memory demand, they do not require algebra routines or the solution of large and complicated systems of nonlinear equations, and they are especially suited for discretizations using the method of lines of two and three dimensional parabolic partial differential equations. In J. Martín-Vaquero and B. Janssen [ Comput. Phys. Commun. 180, No. 10, 1802–1810 (2009; bl 1197.65006)], we showed that previous codes based on stabilized Runge–Kutta algorithms have some difficulties in solving problems with very large eigenvalues and we derived a new code, SERK2, based on sixth-order polynomials. Here, we develop a new method based on second-order polynomials with up to 250 stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs. Numerical experiments with both smooth and nonsmooth data support the efficiency and accuracy of the new algorithms when compared to other well-known second-order methods such as RKC and ROCK2.

References in zbMATH (referenced in 11 articles , 1 standard article )

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  1. Li, Xiao; Ju, Lili; Hoang, Thi-Thao-Phuong: Overlapping domain decomposition based exponential time differencing methods for semilinear parabolic equations (2021)
  2. Asante-Asamani, E. O.; Kleefeld, A.; Wade, B. A.: A second-order exponential time differencing scheme for non-linear reaction-diffusion systems with dimensional splitting (2020)
  3. Tang, Xiao; Xiao, Aiguo: Improved Runge-Kutta-Chebyshev methods (2020)
  4. Martín-Vaquero, J.; Kleefeld, A.: ESERK5: a fifth-order extrapolated stabilized explicit Runge-Kutta method (2019)
  5. Martín-Vaquero, J.; Queiruga-Dios, A.; Martín del Rey, A.; Encinas, A. H.; Hernández Guillén, J. D.; Rodríguez Sánchez, G.: Variable step length algorithms with high-order extrapolated non-standard finite difference schemes for a SEIR model (2018)
  6. Mouloud, A.; Fellouah, H.; Wade, B. A.; Kessal, M.: Time discretization and stability regions for dissipative-dispersive Kuramoto-Sivashinsky equation arising in turbulent gas flow over laminar liquid (2018)
  7. Kleefeld, B.; Martín-Vaquero, J.: SERK2v3: Solving mildly stiff nonlinear partial differential equations (2016)
  8. Martín-Vaquero, J.; Kleefeld, B.: Extrapolated stabilized explicit Runge-Kutta methods (2016)
  9. Bhatt, H. P.; Khaliq, A. Q. M.: The locally extrapolated exponential time differencing LOD scheme for multidimensional reaction-diffusion systems (2015)
  10. Martín-Vaquero, J.; Khaliq, A. Q. M.; Kleefeld, B.: Stabilized explicit Runge-Kutta methods for multi-asset American options (2014)
  11. Kleefeld, B.; Martín-Vaquero, J.: SERK2v2: A new second-order stabilized explicit Runge-Kutta method for stiff problems (2013)