Stability theory of TASE-Runge-Kutta methods with inexact Jacobian
CoRR(2024)
摘要
This paper analyzes the stability of the class of Time-Accurate and
Highly-Stable Explicit Runge-Kutta (TASE-RK) methods, introduced in 2021 by
Bassenne et al. (J. Comput. Phys.) for the numerical solution of stiff Initial
Value Problems (IVPs). Such numerical methods are easy to implement and require
the solution of a limited number of linear systems per step, whose coefficient
matrices involve the exact Jacobian J of the problem. To significantly reduce
the computational cost of TASE-RK methods without altering their consistency
properties, it is possible to replace J with a matrix A (not necessarily
tied to J) in their formulation, for instance fixed for a certain number of
consecutive steps or even constant. However, the stability properties of
TASE-RK methods strongly depend on this choice, and so far have been studied
assuming A=J.
In this manuscript, we theoretically investigate the conditional and
unconditional stability of TASE-RK methods by considering arbitrary A. To
this end, we first split the Jacobian as J=A+B. Then, through the use of
stability diagrams and their connections with the field of values, we analyze
both the case in which A and B are simultaneously diagonalizable and not.
Numerical experiments, conducted on Partial Differential Equations (PDEs)
arising from applications, show the correctness and utility of the theoretical
results derived in the paper, as well as the good stability and efficiency of
TASE-RK methods when A is suitably chosen.
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