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Efficient high-order two-derivative DIRK methods with optimized phase errors

Julius Ehigie, Vu Thai Luan

TL;DR

The article advances time integration for oscillatory and stiff problems by constructing high-order two-derivative DIRK methods with optimized phase errors. It provides a global convergence framework for TDDIRK methods, develops a dispersion/dissipation–driven optimization to design new schemes, and introduces several concrete instances (OTDDIRK4s2a, OTDDIRK4s2b, TDDIRK5s2, OTDDIRK5s3) with demonstrated improvements in stability and phase accuracy. Stability analysis and comprehensive numerical experiments on ODEs and PDEs show that the optimized schemes deliver higher accuracy and efficiency than existing DIRK/TDDIRK schemes, especially for oscillatory problems. The work highlights the practical impact of phase-accurate, high-order TDDIRK time integrators and points to future extensions to parallel and symmetric formulations for nonlinear wave applications.

Abstract

This work constructs and analyzes new efficient high-order two-derivative diagonally implicit Runge--Kutta (TDDIRK) schemes with optimized phase errors. Specifically, we present a convergence result for TDDIRK methods and investigate their optimized phase errors and linear stability analysis. Based on these, we derive new families of 2-stage fourth-order, 2-stage fifth-order, and 3-stage fifth-order TDDIRK schemes. Finally, we provide numerical experiments at both the ODE and PDE levels to demonstrate the accuracy and efficiency of these new schemes compared to known DIRK schemes in the literature.

Efficient high-order two-derivative DIRK methods with optimized phase errors

TL;DR

The article advances time integration for oscillatory and stiff problems by constructing high-order two-derivative DIRK methods with optimized phase errors. It provides a global convergence framework for TDDIRK methods, develops a dispersion/dissipation–driven optimization to design new schemes, and introduces several concrete instances (OTDDIRK4s2a, OTDDIRK4s2b, TDDIRK5s2, OTDDIRK5s3) with demonstrated improvements in stability and phase accuracy. Stability analysis and comprehensive numerical experiments on ODEs and PDEs show that the optimized schemes deliver higher accuracy and efficiency than existing DIRK/TDDIRK schemes, especially for oscillatory problems. The work highlights the practical impact of phase-accurate, high-order TDDIRK time integrators and points to future extensions to parallel and symmetric formulations for nonlinear wave applications.

Abstract

This work constructs and analyzes new efficient high-order two-derivative diagonally implicit Runge--Kutta (TDDIRK) schemes with optimized phase errors. Specifically, we present a convergence result for TDDIRK methods and investigate their optimized phase errors and linear stability analysis. Based on these, we derive new families of 2-stage fourth-order, 2-stage fifth-order, and 3-stage fifth-order TDDIRK schemes. Finally, we provide numerical experiments at both the ODE and PDE levels to demonstrate the accuracy and efficiency of these new schemes compared to known DIRK schemes in the literature.

Paper Structure

This paper contains 15 sections, 2 theorems, 64 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Order conditions and convergence analysis for TDDIRK methods
  3. Order conditions
  4. Global error analysis
  5. Construction of new TDDIRK schemes
  6. Dispersion and dissipation
  7. Optimized two--stage fourth-order schemes
  8. An optimal two-stage fifth-order scheme
  9. Optimized three--stage fifth-order schemes
  10. Stability regions of the newly derived TDDIRK schemes
  11. Numerical experiments
  12. The 2D harmonic oscillator
  13. Advection with a nonlinear source term
  14. 2D advection-diffusion-reaction equation
  15. Conclusion

Key Result

lemma thmcounterlemma

Under the assumptions that $y(t) \in \mathbb{R}^N$ is sufficiently smooth and that $f(y): \mathbb{R}^N\rightarrow \mathbb{R}^N$ is sufficiently differentiable with bounded derivatives and satisfies a local Lipschitz condition, the function $g(y)= f'(y)f(y): \mathbb{R}^N \to \mathbb{R}^N$ is also l for all $y, \ z \in \mathbb{R}^N$.

Figures (7)

  • Figure 1: Stability plots for TDDIRK schemes.
  • Figure 2: Accuracy (left) and efficiency (right) plots of 4th-order schemes for Example 5.1.
  • Figure 3: Accuracy (left) and efficiency (right) plots of 5th-order schemes for Example 5.1.
  • Figure 4: Accuracy (left) and efficiency (right) plots of 4th-order schemes for Example 5.2.
  • Figure 5: Accuracy (left) and efficiency (right) plots of 5th-order schemes for Example 5.2.
  • ...and 2 more figures

Theorems & Definitions (5)

  • lemma thmcounterlemma
  • proof
  • theorem 1
  • proof
  • definition thmcounterdefinition: Dispersion and Dissipation