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Modified Singly-Runge-Kutta-TASE methods for the numerical solution of stiff differential equations

M. Calvo, J. I. Montijano, L. Rández

TL;DR

The paper tackles efficient numerical solution of stiff IVPs by introducing Modified Singly-RKTASE (MSRKTASE), which assigns a distinct Singly-TASE operator per Runge-Kutta stage. By establishing an equivalence to $W$-methods, the authors derive order conditions and stability criteria, enabling the design of MSRKTASE schemes that balance per-stage flexibility with overall accuracy. They construct explicit MSRKTASE2 and MSRKTASE3 methods, achieving improved stability (including $L$-stability) and markedly reduced leading-error coefficients relative to prior Singly-RKTASE schemes. Numerical experiments on linear and nonlinear stiff systems show significant improvements in efficiency and accuracy, with performance competitive against SDIRK methods, especially when LU-factorization costs are a primary concern.

Abstract

Singly-TASE operators for the numerical solution of stiff differential equations were proposed by Calvo et al. in J.Sci. Comput. 2023 to reduce the computational cost of Runge-Kutta-TASE (RKTASE) methods when the involved linear systems are solved by some $LU$ factorization. In this paper we propose a modification of these methods to improve the efficiency by considering different TASE operators for each stage of the Runge-Kutta. We prove that the resulting RKTASE methods are equivalent to $W$-methods (Steihaug and Wolfbrandt, Mathematics of Computation,1979) and this allows us to obtain the order conditions of the proposed Modified Singly-RKTASE methods (MSRKTASE) through the theory developed for the $W$-methods. We construct new MSRKTASE methods of order two and three and demonstrate their effectiveness through numerical experiments on both linear and nonlinear stiff systems. The results show that the MSRKTASE schemes significantly enhance efficiency and accuracy compared to previous Singly-RKTASE schemes.

Modified Singly-Runge-Kutta-TASE methods for the numerical solution of stiff differential equations

TL;DR

The paper tackles efficient numerical solution of stiff IVPs by introducing Modified Singly-RKTASE (MSRKTASE), which assigns a distinct Singly-TASE operator per Runge-Kutta stage. By establishing an equivalence to -methods, the authors derive order conditions and stability criteria, enabling the design of MSRKTASE schemes that balance per-stage flexibility with overall accuracy. They construct explicit MSRKTASE2 and MSRKTASE3 methods, achieving improved stability (including -stability) and markedly reduced leading-error coefficients relative to prior Singly-RKTASE schemes. Numerical experiments on linear and nonlinear stiff systems show significant improvements in efficiency and accuracy, with performance competitive against SDIRK methods, especially when LU-factorization costs are a primary concern.

Abstract

Singly-TASE operators for the numerical solution of stiff differential equations were proposed by Calvo et al. in J.Sci. Comput. 2023 to reduce the computational cost of Runge-Kutta-TASE (RKTASE) methods when the involved linear systems are solved by some factorization. In this paper we propose a modification of these methods to improve the efficiency by considering different TASE operators for each stage of the Runge-Kutta. We prove that the resulting RKTASE methods are equivalent to -methods (Steihaug and Wolfbrandt, Mathematics of Computation,1979) and this allows us to obtain the order conditions of the proposed Modified Singly-RKTASE methods (MSRKTASE) through the theory developed for the -methods. We construct new MSRKTASE methods of order two and three and demonstrate their effectiveness through numerical experiments on both linear and nonlinear stiff systems. The results show that the MSRKTASE schemes significantly enhance efficiency and accuracy compared to previous Singly-RKTASE schemes.
Paper Structure (8 sections, 1 theorem, 43 equations, 4 figures, 3 tables)

This paper contains 8 sections, 1 theorem, 43 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Equivalence between modified Singly-RKTASE and W-methods
  3. Order conditions
  4. Modified Singly-RKTASE methods with orders 2 and 3
  5. Methods with $s=2$, $r=2$ and order 2
  6. Methods with $s=3$, $r=3$ and order 3
  7. Numerical experiments
  8. Conclusions

Key Result

Theorem 3.1

A Modified Singly-RKTASE method can not have order $r+1$.

Figures (4)

  • Figure 1: Stability regions of the new methods (left). On the right, a zoom of the region near the origin.
  • Figure 2: Efficiency plot for Diffusion problem
  • Figure 3: Efficiency plot for Burger's problem integrated with $W$ the Jacobian matrix evaluated at every step (left) and $W$ the Jacobian matrix evaluated just at the initial time step (right)
  • Figure 4: Efficiency plot for Burger's problem integrated with $W$ equal to the matrix of diffusion term

Theorems & Definitions (1)

  • Theorem 3.1