TY - JOUR
T1 - Stabilized time-series expansions for high-order finite element solutions of partial differential equations
AU - Deeb, Ahmad
AU - Dutykh, Denys
N1 - Publisher Copyright:
© 2024 Wiley Periodicals LLC.
PY - 2024/8
Y1 - 2024/8
N2 - Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time-Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the application of high-order Finite Element (FE) to certain classes of PDEs, such as diffusion equations and the Navier–Stokes (NS) equations, often leads to numerical instabilities. These instabilities limit the number of valid terms in the series, though the efficiency of time-series integration even when resummation techniques like the Borel–Padé–Laplace (BPL) integrators are employed. In this study, we introduce a novel variational formulation for computing the terms of a TSE associated with a given PDE using higher-order FEs. Our approach involves the incorporation of artificial diffusion terms on the left-hand side of the equations corresponding to each power in the series, serving as a stabilization technique. We demonstrate that this method can be interpreted as a minimization of an energy functional, wherein the total variations of the unknowns are considered. Furthermore, we establish that the coefficients of the artificial diffusion for each term in the series obey a recurrence relation, which can be determined by minimizing the condition number of the associated linear system. We highlight the link between the proposed technique and the Discrete Maximum Principle (DMP) of the heat equation. We show, via numerical experiments, how the proposed technique allows having additional valid terms of the series that will be substantial in enlarging the stability domain of the BPL integrators.
AB - Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time-Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the application of high-order Finite Element (FE) to certain classes of PDEs, such as diffusion equations and the Navier–Stokes (NS) equations, often leads to numerical instabilities. These instabilities limit the number of valid terms in the series, though the efficiency of time-series integration even when resummation techniques like the Borel–Padé–Laplace (BPL) integrators are employed. In this study, we introduce a novel variational formulation for computing the terms of a TSE associated with a given PDE using higher-order FEs. Our approach involves the incorporation of artificial diffusion terms on the left-hand side of the equations corresponding to each power in the series, serving as a stabilization technique. We demonstrate that this method can be interpreted as a minimization of an energy functional, wherein the total variations of the unknowns are considered. Furthermore, we establish that the coefficients of the artificial diffusion for each term in the series obey a recurrence relation, which can be determined by minimizing the condition number of the associated linear system. We highlight the link between the proposed technique and the Discrete Maximum Principle (DMP) of the heat equation. We show, via numerical experiments, how the proposed technique allows having additional valid terms of the series that will be substantial in enlarging the stability domain of the BPL integrators.
KW - divergent series resummation
KW - finite element method
KW - stabilization technique
KW - time integration
KW - time-series expansion
UR - http://www.scopus.com/inward/record.url?scp=85194701261&partnerID=8YFLogxK
U2 - 10.1111/sapm.12708
DO - 10.1111/sapm.12708
M3 - Article
AN - SCOPUS:85194701261
SN - 0022-2526
VL - 153
JO - Studies in Applied Mathematics
JF - Studies in Applied Mathematics
IS - 2
M1 - e12708
ER -