Fast explicit time integration schemes for parabolic problems in mechanics

• Published in: International journal for numerical methods in engineering Vol. 125; no. 13
• Main Authors: Oñate, Eugenio, Zárate, Francisco, Gimenez, Juan M., Lohner, Rainald, Idelsohn, Sergio R.
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 15-07-2024; Wiley Subscription Services, Inc
• Subjects: Accuracy; Conduction heating; Conductive heat transfer; explicit time integration schemes; fast explicit time integration; FIC; Finite element method; finite increment calculus; Mathematical analysis; Mechanics; Numerical methods; parabolic equation; Runge-Kutta method; Solid mechanics; Time integration; Transient heat conduction
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.7477

We present a family of fast explicit time integration schemes of first, second and third order accuracy for parabolic problems in mechanics solved via standard numerical methods that have considerable higher computational efficiency versus existing explicit methods of the same order. The derivation of the new explicit schemes is inspired on the finite increment calculus (FIC) procedure used for obtaining stabilized numerical schemes in fluid and solid mechanics. The new (so‐called) explicit FIC‐Time (EFT) schemes allow considerable larger time steps than the standard first order forward Euler (FE) scheme and the second and third order Adams–Bashforth schemes. The comparison with Runge–Kutta schemes also favors the FIC‐Time schemes in terms of the limit time step size (for second order schemes) and the total number of matrix‐vector multiplications per time step (for second and third order schemes). The new first order explicit schemes have a faster convergence to steady‐state than the FE scheme. The accuracy and efficiency of the new EFT schemes are verified in examples of application to the transient heat conduction equation using the finite element method.