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

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.

💡 Research Summary

The manuscript presents the development and analysis of new high‑order two‑derivative diagonally implicit Runge–Kutta (TDDIRK) schemes whose phase errors are explicitly optimized. The authors begin by formulating the initial‑value problem y′=f(y) and introducing the second derivative g(y)=f′(y)f(y). A general s‑stage TDDIRK method is defined by equations (1.2), where the stage values Y_i depend on the current solution y_n, the first derivative f(y_n), and the second derivative g(Y_j). This framework extends classical DIRK methods by incorporating second‑derivative information, which enables higher accuracy and improved dispersion/dissipation characteristics.

Section 2 derives the order conditions for TDDIRK methods up to sixth order (Table 2.1) and provides a global convergence theorem. Lemma 2.1 shows that g is locally Lipschitz under the standard smoothness assumptions on f, and Theorem 2.1 proves that a TDDIRK method satisfying the order conditions of order p (2 ≤ p ≤ 6) converges with global error O(h^p) provided the stepsize h obeys h < 1/(p L_g max|a_{jj}|). The proof proceeds by bounding the stage errors E_i recursively and applying a discrete Grönwall inequality.

Section 3 is the core of the paper: the authors design families of schemes with optimized phase (dispersion) and amplitude (dissipation) errors. Using the linear oscillatory test equation y′=i ω y, they derive the stability function R(iν) for a generic TDDIRK method (eq. 3.4). The phase‑lag Ψ(ν)=ν−arg R(iν) and the dissipation Φ(ν)=1−|R(iν)| are expanded in powers of ν. The goal is to make the leading coefficients of Ψ and Φ as small as possible, ideally zero, which corresponds to higher dispersion and dissipation orders.

For a two‑stage, fourth‑order method they introduce two free parameters α and β (c_1=α, a_{21}=β). Solving the order‑four conditions together with the dispersion‑order‑six / dissipation‑order‑seven requirements yields a unique pair (α,β) that eliminates the ν^5 term in Ψ and the ν^6 term in Φ. This results in the scheme OTDDIRK4s2a, which attains dispersion order 6 and dissipation order 7. A second set of parameters is obtained by targeting dispersion order 8 and dissipation order 5, giving a different fourth‑order scheme with a slightly different error balance.

Analogous procedures are applied to construct two‑stage and three‑stage fifth‑order families. The authors again solve the higher‑order conditions together with phase‑error constraints, producing schemes that achieve dispersion orders up to eight and dissipation orders up to seven while maintaining fifth‑order accuracy.

Section 4 conducts a linear stability analysis. The authors compute the stability regions in the complex plane for the newly derived schemes and compare them with well‑known DIRK methods such as SDIRK4 and SDIRK5. The TDDIRK families exhibit larger stability domains, especially along the negative real axis, indicating better suitability for stiff problems. Moreover, the inclusion of second‑derivative terms expands the admissible stepsize for oscillatory problems without sacrificing stability.

Section 5 presents extensive numerical experiments. First, a suite of ODE tests (linear harmonic oscillator, stiff Van der Pol, nonlinear oscillators) demonstrates that the optimized TDDIRK methods achieve the predicted convergence rates and exhibit markedly smaller global errors than comparable DIRK schemes at the same stepsize. Runtime measurements show that, despite the extra evaluation of g, the overall computational cost is reduced because larger stepsizes can be used while preserving accuracy.

Second, PDE tests are performed after spatial discretization of wave‑type equations (1‑D wave equation, 2‑D Schrödinger equation) and of a nonlinear Klein‑Gordon equation. Long‑time integration highlights the accumulation of phase errors in standard DIRK methods, leading to noticeable phase drift and energy loss. In contrast, the OTDDIRK schemes maintain phase fidelity over thousands of periods; the energy error remains two orders of magnitude lower. The authors also report that the dispersion‑optimized schemes allow coarser spatial meshes without sacrificing the overall solution quality.

The paper concludes that incorporating second‑derivative information into diagonally implicit Runge–Kutta frameworks yields a powerful class of integrators that combine high order, enlarged stability regions, and controllable phase/dissipation errors. The authors suggest future work on multi‑stage, multi‑parameter automatic optimization, adaptive stepsize control based on phase‑error estimators, and extensions to exponential‑type TDDIRK methods for highly oscillatory problems. Potential applications include electromagnetic wave propagation, fluid‑structure interaction, and quantum dynamics where long‑time phase accuracy is critical.