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Order elevation of directly self-starting sub-step implicit integrators for transient dynamics

Jinze Li, Yaokun Liu, Kewei Chen, Hua Li, Kaiping Yu

arXiv:2609.10240v1math.NAcs.NAphysics.comp-ph

TL;DR

Most existing directly self-starting sub-step methods fix the final stage at the time-interval endpoint, potentially restricting attainable order. This study frees the stage locations and constructs two method families, reaching seventh order with six sub-steps while retaining controllable dissipation in the sth-order family.

  • Problem

    Most existing sub-step methods fix the last stage at the interval endpoint, a constraint that may restrict attainable temporal accuracy for a given number of sub-steps.

    Source: §1.3 · p. 3

  • Method

    The study generalizes directly self-starting sub-step integration by treating stage locations as design variables and constructing accuracy–dissipation variants for first- and second-order systems.

    Source: §6 · pp. 48–49

  • Results

    With up to six sub-steps, the framework reaches seventh-order accuracy; sth-order members retain user-controllable dissipation, while (s+1)th-order members have fixed dissipation.

    Source: §6 · p. 49

  • Takeaways & Limitations

    In undamped systems, odd-sub-step sth-order methods show one-order phase superconvergence and even-sub-step methods show one-order amplitude superconvergence.

    Source: §6 · p. 49

    The fifth-order four-sub-step method is only A(α)-stable, and weak narrow-band amplification means r = 1/2 does not necessarily minimize response error.

    Source: §3.4.1 · p. 24, The four-sub-step implicit method · p. 44

Abstract

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Directly self-starting implicit methods are attractive for transient analysis because they avoid auxiliary starting procedures while retaining the original first- or second-order governing equations. However, most existing formulations usually fix the last sub-step at the end of each time interval, which restricts the attainable order. This study develops a generalized $s$-sub-step implicit framework by releasing this constraint and treating all sub-step locations as design variables. The resulting methods admit a unified Runge--Kutta representation for both first- and second-order transient systems and preserve identical effective matrices over all sub-steps. Accuracy conditions are derived by simultaneously matching the numerical amplification factor and load operator, thereby accounting for both homogeneous and forced responses. For $s=1,~\cdots,~6$, two complementary families are obtained: $s$th-order members with user-controllable high-frequency numerical dissipation and adjustable sub-step locations, and $(s+1)$th-order members obtained by selecting the sub-step locations, with fixed dissipation. The latter reach up to seventh-order accuracy without increasing the number of sub-steps, although some high-order members are $A(\alpha)$-stable with stability angles extremely close to $90^\circ$. Analytical amplitude and phase errors further reveal parity-dependent superconvergence in undamped systems, and appropriate parameter selections can substantially increase either phase or amplitude accuracy beyond the formal order. Numerical benchmarks confirm the predicted convergence orders and the controllable suppression of spurious high-frequency responses.

1 Introduction

Implicit integration offers improved stability for transient dynamics, while conventional sub-step methods largely remain second-order. This study relaxes the final-stage constraint to expand the accuracy available within a fixed number of sub-steps.

1 Introduction

  • Implicit methods permit time increments selected for accuracy rather than stability, but their algebraic solves contrast with the time-step restrictions of explicit methods.Large-scale, nonlinear, and long-duration simulations motivate robust and efficient schemes.Source: §1 · p. 2

1.1 Second-order accurate methods

  • Sub-step methods enlarge the design space while retaining direct integration, yet most existing algorithms remain second-order despite growing demand for accurate long-term simulations.Identical effective matrices can avoid repeated factorizations and support spectral properties such as minimal period elongation and suppression of spurious high-frequency responses.Source: §1.1 · p. 2

1.2 High-order accurate methods

  • High-order routes based on first-order reformulations can increase system dimensions and computational cost, while directly self-starting sub-step methods retain the original dynamics and their matrix structure.Prior directly self-starting methods can suffer order reduction for forced vibrations; later work and flexible stage distributions further developed higher-order sub-step schemes.Source: §1.2 · p. 3

1.3 Motivation and contributions

  • Fixing the last sub-step at the interval endpoint may restrict attainable accuracy, so this study makes all sub-step locations design variables while preserving direct starting and identical effective matrices.For a given s, the framework trades controllable dissipation in an sth-order member against fixed dissipation in an (s+1)th-order member; analysis also finds parity-dependent superconvergence and near-90° A(α)-stability in some higher-order methods.Source: §1.3 · pp. 3–4

2 The directly self-starting sub-step implicit framework

The framework formulates sub-step integration for first- and second-order transient systems while retaining direct starting. It enforces shared effective matrices and derives accuracy and stability conditions, including control of high-frequency dissipation.

2 The directly self-starting sub-step implicit framework

  • Sub-step locations define stages for second-order dynamics, and the method directly starts without requiring the initial acceleration; the first-order formulation likewise avoids requiring the initial derivative.Both formulations admit a unified Runge–Kutta representation, without converting second-order dynamics into an equivalent first-order system.Source: §2 · pp. 4–5

2.1 Identical effective matrices

  • Enforcing identical effective matrices reduces the formulation to an SDIRK method and lets linear constant-step problems reuse one assembled and factorized matrix across sub-steps.This avoids repeated assembly and factorization, particularly relevant to large-scale models.Source: §2.1 · pp. 5–6

2.2 Accuracy

  • Accuracy conditions match both the numerical amplification factor and load operator, accounting for homogeneous and forced contributions and addressing order reduction associated with load approximation.Consistency also requires each sub-step coefficient to match its corresponding temporal location.Source: §2.2 · pp. 7–8

2.3 Stability

  • Stability is analyzed through the Runge–Kutta stability function by checking its poles and the sign of the associated E(y) polynomial.The framework applies established Runge–Kutta stability criteria to the proposed methods.Source: §2.3 · p. 8

2.4 Dissipation control

  • High-frequency dissipation is prescribed through ρ∞, which determines the first sub-step size γ1 and can also quantify dissipation once the other parameters are fixed.Damping spurious high-frequency modes is motivated by their potential to impair solution accuracy and stability after spatial discretization.Source: §2.4 · pp. 8–9

3 Developments and analyses

The framework uses free sub-step locations to construct two accuracy–dissipation configurations for a fixed number of sub-steps.

3 Developments and analyses

  • For s sub-steps, the framework offers sth-order accuracy with controllable high-frequency dissipation or (s+1)th-order accuracy with fixed dissipation by using an additional order condition.The higher-order configuration requires no additional sub-steps; the sth-order configuration retains adjustable locations, with γ1 associated with ρ∞.Source: §3 · p. 9

3.1 Single sub-step: s = 1

The single-sub-step case recovers a previously developed implicit method and provides a reference for the framework. Its accuracy, amplitude and phase errors, and stability vary with the dissipation parameter.

3.1 Single sub-step: s = 1

  • The single-sub-step formulation uses the consistency condition α11 = γ1, and first-order accuracy requires β1 = 1.The method is presented as a reference for relating the generalized framework to classical implicit algorithms.Source: §3.1 · pp. 9–10
  • For ρ∞ in (−1, 1), the single-sub-step method is first-order accurate and unconditionally stable; at ρ∞ = 1, it is second-order and non-dissipative for undamped systems.The authors note that this single-sub-step algorithm was developed previously.Source: §3.1 · p. 10
  • In the undamped first-order member, phase error is O(∆t^2) while amplitude error is O(∆t); the undamped second-order member has zero amplitude decay and O(∆t^2) phase error.For damped systems, both errors are O(∆t) in the first-order member and O(∆t^2) in the second-order member.Source: §3.1 · p. 10
  • All tested ρ∞ cases contain the left half-plane in their stability regions, while decreasing ρ∞ reduces high-frequency spectral radius and strengthens dissipation.At ρ∞ = 1, |R(∞)| = 1, corresponding to the non-dissipative case.Source: §3.1 · p. 10

3.2 Two sub-steps: s = 2

The two-sub-step formulation frees the second sub-step location while retaining identical effective matrices, allowing higher order without adding sub-steps. It provides a second-order family with controllable high-frequency dissipation and a third-order member with fixed dissipation; both retain unconditional stability.

3.2 Two sub-steps: s = 2

  • Unlike earlier methods that fix γ2 = 1, the formulation treats the second sub-step location as independent, and admissible γ2 choices do not affect stability or spectral properties.The two-sub-step method also uses identical effective matrices across its sub-steps.Source: §3.2 · pp. 10, 11, 13

3.2.1 Higher-order accuracy

  • The third-order two-sub-step member achieves higher accuracy without compromising unconditional stability, with its added order condition fixing high-frequency dissipation.The second-order and third-order algorithms share the same sub-step splitting strategy.Source: §3.2.1 · pp. 13–14

3.2.2 Spectral properties

  • The second-order family lets users control high-frequency dissipation through ρ∞ while maintaining near-unit spectral radius and negligible artificial damping at low frequencies.Smaller ρ∞ produces stronger high-frequency damping.Source: §3.2.2 · p. 15
  • The recommended γ1 branch generally gives smaller period errors than the alternative branch, and the third-order member further improves low-frequency phase accuracy.The improvement is attributed to the additional order condition and retains the stability demonstrated for that member.Source: §3.2.2 · p. 15
  • For undamped systems, the second-order member has third-order amplitude accuracy, while the third-order member has fourth-order phase accuracy.In the damped case, both amplitude and phase errors follow their respective formal orders.Source: §3.2.2 · pp. 15–16
  • For the non-dissipative cases γ1 = 1/2 or 1/4, the amplitude error vanishes identically.These choices correspond to ρ∞ = 1.Source: §3.2.2 · p. 16

3.3 Three sub-step: s = 3

The three-sub-step framework derives algorithmic parameters from accuracy and stability conditions, yielding third-order methods with controllable dissipation and fourth-order methods with fixed dissipation. Comparisons show improved phase accuracy over a second-order method at the same three-stage complexity, alongside tunable mid- and high-frequency damping.

3.3 Three sub-step: s = 3

  • The three-sub-step method reaches third-order accuracy when its amplification-factor and load-operator error conditions are simultaneously satisfied.The conditions are c1 = c2 = c3 = c4 = 0.Source: §3.3 · p. 17
  • All considered three-sub-step cases are unconditionally stable, including the fourth-order member, although some regions extend into the right half-plane.The left half-plane remains included in the stability region.Source: §3.3 · p. 18, Figure 6 · p. 19, §3.3.1 · p. 20

3.3.1 Higher-order accuracy

  • For arbitrary user-specified ρ∞, the framework provides third-order accuracy with controllable dissipation; a selected ρ∞ yields fourth-order accuracy with fixed |ρ∞| ≈ 0.63041.The two configurations use the same sub-step splitting strategy.Source: §3.3.1 · p. 20

3.3.2 Spectral properties

  • For undamped systems, the third-order member has one-order phase superconvergence, while the fourth-order member has one-order amplitude superconvergence.The fourth-order member's amplitude error is O(∆t5) and its phase error remains O(∆t4).Source: §3.3.2 · p. 21
  • The phase-optimized third-order member achieves sixth-order phase accuracy for undamped problems while retaining unconditional stability and fixed high-frequency dissipation.Its recommended γ1 is 0.9756745886944403, corresponding to ρ∞ ≈ −0.67851.Source: §3.3.2 · p. 22

3.3.3 Comparisons with existing three-sub-step implicit methods

  • Compared with the ρ∞-MSSBN3 method, the proposed methods provide stronger mid- and high-frequency dissipation; the Frutos–Serna method is non-dissipative across the frequency range.The proposed methods' damping levels are controllable through ρ∞.Source: Figure 8 · p. 22, §3.3.3 · p. 23
  • The proposed third- and fourth-order methods substantially outperform the second-order ρ∞-MSSBN3 method in low- and mid-frequency period error without increasing computational complexity.The phase-optimized third-order member has the smallest undamped period error over most of the range; the fourth-order member is best among those compared in the damped case.Source: Figure 8 · p. 22, §3.3.3 · p. 23

3.4 More sub-steps: s ≥4

For four to six sub-steps, the framework gives either sth-order methods with controllable high-frequency dissipation or (s + 1)th-order methods with fixed dissipation. The additional order condition fixes the sub-step locations, and the higher-order members reach seventh order for six sub-steps.

3.4.1 Four sub-steps: s = 4

  • For four sub-steps, fourth-order members are unconditionally stable with controllable dissipation, while the fifth-order member has slight mid-frequency amplification and is only A(α)-stable.Its stability angle is approximately 89.99173663°; the four-step method’s undamped amplitude error can be improved from O(∆t^5) to O(∆t^7) with a suitable γ1.Source: §3.4.1 · pp. 23–25

3.4.2 Five sub-steps: s = 5

  • For five sub-steps, fifth-order members allow dissipation control, whereas the sixth-order member fixes dissipation and improves amplitude and phase accuracy, especially for damped problems.The sixth-order member with γ1 = 0.47327 is A-stable and has smaller period errors than the A(α)-stable sixth-order alternative; the fifth-order member can reach O(∆t^8) phase accuracy when undamped.Source: §3.4.2 · pp. 25–27

3.4.3 Six sub-steps: s = 6

  • For six sub-steps, the sixth-order members offer controllable dissipation, while the seventh-order members fix it; the preferred seventh-order choice has only a barely perceptible spectral-radius overshoot.The preferred member has γ1 = 0.55670 and spectral radius approximately 0.72080 at high frequency; an alternative seventh-order choice has a more pronounced mid-frequency overshoot.Source: §3.4.3 · pp. 27–28
  • The preferred seventh-order six-sub-step member has phase errors smaller than the alternative seventh-order member and stability close to unconditional, but its stability is A(α), not unconditional.Its high-frequency spectral radius is approximately 0.72080; the alternative has a more pronounced mid-frequency amplification and inferior period error.Source: §3.4.3 · pp. 27–28

3.4 More sub-steps: s ≥4

  • For the six-sub-step implicit method, the sixth-order member’s undamped amplitude error improves from its formal O(∆t6) to O(∆t7), while the seventh-order member’s undamped phase error is O(∆t8).For the sixth-order member, selecting γ1 = 0.5409068780733081 eliminates the ∆t7 amplitude-error term and yields O(∆t9) amplitude error for undamped problems; this value satisfies unconditional stability.Source: §3.4.3 · p. 29

3.5 Recovery of additional quantities

The methods advance the primary state variables directly, without requiring highest derivatives as integration inputs. Additional response quantities can be recovered afterward when needed.

3.5 Recovery of additional quantities

  • For second-order dynamics, the methods update displacement and velocity without explicitly supplying acceleration; first-order methods likewise advance state without outputting time derivatives.Acceleration or derivative responses require additional procedures only when users want those quantities as outputs.Source: §3.5 · pp. 29–30
  • Postprocessing for additional response quantities does not affect the methods’ integration accuracy, stability, spectral properties, or dissipation.The cited recovery approaches include finite-difference postprocessing and reconstruction from governing equations.Source: §3.5 · p. 30

4 Comparisons

The framework trades dissipation control for an extra formal order when all sub-step locations are fixed. Analytical errors and spectral comparisons show that parameter choices can further improve accuracy beyond formal order.

4 Comparisons

  • Undamped superconvergence alternates with sub-step parity: odd-step methods gain phase order, while even-step methods gain amplitude order.Appropriate parameter choices can improve these errors by three orders beyond formal accuracy for the specified odd- and even-step methods.Source: §4 · pp. 30–31
  • The optimized sth-order methods remain unconditionally stable across the plotted frequency range, and their parameter choices balance damping with low-frequency accuracy rather than maximizing dissipation.The spectral comparisons show theoretical error-order slopes for damped cases and parity-dependent superconvergence for undamped cases.Source: §4 · pp. 30–31
  • For undamped systems, optimized parameters can raise phase accuracy to sixth and eighth order for three- and five-step methods, and amplitude accuracy to seventh and ninth order for four- and six-step methods.The two-step method is non-dissipative at γ1 = 1/2 or 1/4, with vanishing undamped amplitude error.Source: §4 · p. 31
  • Some higher-order members are only A(α)-stable, although their reported stability angles remain extremely close to 90°.The higher-order construction increases accuracy by one order but removes independent control of high-frequency dissipation.Source: §4 · p. 32
  • The formal-order methods let ρ∞ control high-frequency dissipation, while the (s + 1)th-order methods fix dissipation by consuming the sub-step freedom.The dissipation parameter spans ρ∞ ∈ [0, 1], and some formulations also allow negative values.Source: §4 · pp. 30–32

5 Numerical examples

The numerical examples test second-order structural dynamics, while the framework’s theoretical results also apply to first-order systems. In the damped, forced oscillator, observed convergence confirms the predicted orders for both adjustable-dissipation and fixed-dissipation methods.

5 Numerical examples

  • The numerical investigations focus on second-order structural dynamics and compare members within the proposed framework rather than comprehensively benchmarking existing methods.The shared Butcher tableau and accuracy, stability, and dissipation conditions make the theoretical results applicable to first-order transient problems as well.Source: §5 · pp. 32–33

5.1 The damped and forced oscillator

  • The damped, forced oscillator confirms sth-order convergence for displacement and velocity across s = 1–6 and ρ∞ = 0, 0.5, and 1.The convergence rates are almost unaffected by ρ∞, which changes high-frequency spectral behavior without changing formal order except for s = 1.Source: §5.1 · p. 33
  • The fixed-parameter (s + 1)th-order members converge at their predicted orders, from third through seventh order for two through six sub-steps.An apparent fifth-order slope for the fourth-order three-sub-step member is attributed to a small error coefficient in this oscillator; its asymptotic order remains fourth.Source: §5.1 · p. 33

5.2 The model problem with stiff and flexible components

The stiff-flexible model tests whether the proposed methods can suppress high-frequency spurious responses while preserving low-frequency physical behavior. The results show improved accuracy with higher-order methods and sub-step counts, alongside a computational-cost trade-off.

5.2 The model problem with stiff and flexible components

  • The benchmark contains a dominant low-frequency physical mode and a high-frequency component caused by a stiff spring, making it suitable for testing spurious-response suppression.The approximate modal frequencies are 1.0 rad/s and 3162.3 rad/s, respectively.Source: §5.2 · pp. 33, 35, Figure 18 · p. 35
  • For the single-sub-step method, ρ∞ = 0 eliminates the high-frequency oscillation while preserving the low-frequency response; increasing ρ∞ retains more high-frequency information.At ρ∞ = 1, the method is non-dissipative and the spurious oscillation remains visible; this case is second-order accurate.Source: §5.2 · p. 35
  • With ρ∞ = 0, the sth-order methods accurately capture low-frequency responses, remove high-frequency oscillations, and show continuously decreasing errors as sub-step count increases.The methods are compared at the same high-frequency dissipation level.Source: §5.2 · p. 35
  • Almost all (s + 1)th-order methods capture the low-frequency response and suppress spurious oscillations, but five- and six-sub-step methods show larger node-2 velocity amplitude deviations.The deviations are associated with sub-step sizes selected to satisfy additional accuracy constraints, which leave fewer parameters for optimizing spectral properties such as high-frequency dissipation.Source: §5.2 · p. 35
  • The (s + 1)th-order methods generally yield smaller errors than their sth-order counterparts, and the six-sub-step method achieves the smallest overall errors.The comparison is based on displacement and velocity errors at node 3.Source: §5.2 · p. 36
  • At the same time step size, increasing the number of sub-steps improves accuracy but raises computational cost through additional intermediate evaluations.The paper frames sub-step selection as a balance between accuracy and computational efficiency.Source: §5.2 · p. 36
  • The proposed three-sub-step methods have smaller numerical errors than two existing three-sub-step algorithms while effectively attenuating the high-frequency spurious mode.The comparison uses absolute displacement and velocity errors at node 3.Source: §5.2 · p. 36

5.3 An elastic bar with axial tip loading

This section examines elastic-bar time-integration accuracy in relation to spatial-discretization error, using linear and quadratic finite elements and mass weighting. It considers how CFL selection and optimized mass weighting affect dispersion accuracy, including the influence of weak amplitude amplification on the practical optimum.

The single-sub-step implicit method

  • For s = 1, the optimal CFL can cancel leading dispersion error, but very small steps may expose stronger spurious oscillations; dissipation can also mask the optimal-CFL benefit.With ρ∞ = 1, the error minimum occurs around the optimal CFL; with ρ∞ = 0.8, strong algorithmic dissipation also attenuates physical waves, and CFL = 0.4 appears slightly more accurate than the theoretical optimum.Source: The single-sub-step implicit method · pp. 39–40, Figure 25 · p. 40, Figure 24 · p. 40

The two-sub-step implicit method

  • For s = 2, the second-order method benefits from mass-dependent optimal CFL values, while the third-order method reaches O(ω5) dispersion with r = 1/2 but has no real optimal CFL.For the second-order method, CFLOpt ≈ 1.3154 at r = 1 and ≈ 0.0221 at r = 0; for the third-order method, r = 0 or 1 leaves O(ω3) error.Source: The two-sub-step implicit method · pp. 40–41, Figure 26 · p. 41

The three-sub-step implicit method

  • For s = 3, r = 1/2 and an optimal CFL balance O(ω5) errors to achieve O(ω7), while raising temporal order from third to fourth does not improve attainable dispersion with linear elements.The optimal CFL reduces oscillations; both third- and fourth-order methods remain limited by spatial discretization unless the spatial scheme or mass matrix is improved.Source: The three-sub-step implicit method · p. 42, Figure 28 · p. 43

The four-sub-step implicit method

  • For s = 4, the fourth-order method with r = 1/2 and an optimal CFL improves dispersion from O(ω5) to O(ω7), but fifth-order performance is shaped by stability as well as dispersion.The fifth-order method has no real optimal CFL, and weak amplification of some frequencies means r = 1/2 need not minimize the complete response error.Source: The four-sub-step implicit method · pp. 43–44

The five- and six-sub-step implicit methods

  • For s = 5 and 6, leading dispersion errors depend only on spatial discretization, so further temporal-order increases or sub-step tuning cannot improve accuracy without better spatial discretization.No additional numerical validation is reported because spatial error dominates.Source: The five- and six-sub-step implicit… · p. 44

5.3.2 Spatial discretization with three-node quadratic elements

  • Using 100 three-node quadratic elements eliminates the third-order spatial dispersion error present with linear elements, enabling assessment of higher-order time-integration behavior.The quadratic-element formulation also uses a mass-weighting parameter r.Source: §5.3.2 · pp. 44–45

The single- and two-sub-step implicit methods

  • With quadratic elements, the single-sub-step method cannot eliminate its leading temporal dispersion error using real parameter values, while the second-order two-sub-step method has no theoretical optimal CFL.An empirical region near CFL ≈ 0.25 minimizes oscillations for the second-order method, but this is not dispersion-error cancellation; the third-order member can balance spatial and temporal errors using an optimal CFL.Source: The single- and two-sub-step implicit… · p. 45
  • With quadratic finite elements, the third-order two-sub-step method can exploit its higher temporal accuracy through CFL optimization, whereas the second-order member lacks that theoretical mechanism.The observed optimum for the second-order member mainly redistributes high-frequency errors rather than canceling the leading dispersion term.Source: The single- and two-sub-step implicit… · p. 45

The three-sub-step implicit method

  • With quadratic elements, three-sub-step methods balance dispersion errors using an optimal CFL when ϕ2 ≠ 0; the fourth-order member shares the third-order member’s optimum.Quadratic elements remove third-order spatial dispersion error, leaving leading frequency error O(ω5). When ϕ2 = 0, choosing r = 1/3 removes the fifth-order spatial error and improves leading accuracy to O(ω7).Source: The three-sub-step implicit method · p. 46, Figure 31 · p. 47

The four-sub-step implicit method

  • For the fourth-order four-sub-step method, choosing r < 1/3 and the optimal CFL eliminates the leading ω5 term and improves numerical frequency accuracy to O(ω7).For ρ∞ = 0.8 and r = 0, the reported CFLOpt is 0.6021; this choice improves agreement near wave fronts versus CFL = 0.4 and 0.8.Source: The four-sub-step implicit method · pp. 46–47, Figure 32 · p. 48
  • The fifth-order four-sub-step method reaches O(ω9) dispersion accuracy only with r = 1/3 and an optimal CFL number, but its practical minimum error occurs near CFL = 0.177.A narrow frequency range of weak numerical amplification under A(α)-stability shifts the total-error minimum from the theoretical phase-based optimum.Source: The four-sub-step implicit method · pp. 47–48, Figure 32 · p. 48

The five- and six-sub-step implicit methods

  • CFLOpt = 0.7090 for the six-sub-step method with ρ∞ = 0.8 and r = 1/3, while the strongly dissipative case has CFLOpt = 0.5817 and suppresses wave-front oscillations.The five- and six-sub-step methods require r = 1/3 to eliminate the leading ω5 spatial error; an optimal CFL number then balances the remaining ω7 terms. The six-sub-step method is shown as representative of both.Source: The five- and six-sub-step implicit… · p. 48

6 Conclusions

The study generalizes directly self-starting s-sub-step integration by making sub-step locations design variables, with a unified Runge–Kutta form and identical effective matrices. It constructs higher-order families, analyzes their stability and spectral accuracy, and benchmarks their convergence and computational trade-offs.

6 Conclusions

  • The generalized directly self-starting framework treats sub-step locations as design variables and retains a unified Runge–Kutta representation with identical effective matrices.It applies to both first- and second-order transient problems while operating on the original governing equations.Source: §6 · pp. 48–49
  • The framework constructs sth-order methods with controllable high-frequency dissipation and (s + 1)th-order methods with fixed dissipation, reaching seventh order at s = 6.The higher-order family uses the available sub-step locations to satisfy an additional order condition without increasing the number of sub-steps.Source: §6 · p. 49
  • Some (s + 1)th-order members are A(α)-stable rather than strictly unconditionally stable, with stability angles extremely close to 90°.The sth-order members preserve unconditional stability across the specified ρ∞ range.Source: §6 · p. 49
  • In undamped systems, odd-sub-step sth-order methods have one-order phase superconvergence, while even-sub-step methods have one-order amplitude superconvergence.Optimizing γ1 raises phase accuracy three orders beyond formal order for three- and five-sub-step methods, and amplitude accuracy three orders for four- and six-sub-step methods.Source: §6 · p. 49
  • With linear finite elements, spatial dispersion can mask high-order temporal accuracy, so reducing the time step alone need not approach the exact PDE solution.Appropriate CFL numbers can balance spatial and temporal dispersion; quadratic elements reduce spatial error and expose temporal accuracy more fully.Source: §6 · p. 49
  • Numerical examples confirm the predicted sth- and (s + 1)th-order convergence and suppression of spurious high-frequency components while preserving low-frequency responses.More sub-steps generally improve accuracy at the same time step but increase computational cost, so their number should reflect the accuracy-efficiency balance.Source: §6 · p. 50

A.1 Four sub-steps: s = 4

The four-sub-step appendix relates fourth-order accuracy to a user-selected dissipation parameter and considers fifth-order conditions. No strictly unconditionally stable fifth-order member is found; the practical choice is A(α)-stable with an angle near 90°.

A.1 Four sub-steps: s = 4

  • The fourth-order method relates γ1 to the user-specified ρ∞, while the remaining sub-step sizes remain free subject to stability constraints.For a prescribed ρ∞, the admissible γ1 solution is selected from candidate roots by enforcing unconditional stability.Source: §A.1 · p. 51
  • Fifth-order conditions require local truncation errors of order O(∆t^6), but none of the resulting γ1 candidates satisfies unconditional stability.The adopted practical choice, γ1 ≈ 1.3453664197803336, yields A(α)-stability with α ≈ 89.99173663°.Source: §A.1 · pp. 51–52

A.2 Five sub-steps: s = 5

For five sub-steps, the method supports fifth-order accuracy with controllable dissipation across the full ρ∞ range, or sixth-order accuracy at a specific sub-step choice with fixed dissipation. An alternate sixth-order solution has less favorable period error and stability.

A.2 Five sub-steps: s = 5

  • The fifth-order five-sub-step method remains unconditionally stable for ρ∞ across [−1, 1], preserving controllable high-frequency dissipation.The first sub-step size is set through the dissipation-control condition, with admissible solutions selected to satisfy stability.Source: §A.2 · p. 53
  • Choosing γ1 = 0.4732683912582953 yields a sixth-order method with fixed |ρ∞| ≈ 0.83733, rather than independent dissipation control.The appendix contrasts this with the fifth-order configuration, which permits arbitrary appropriate ρ∞ in [−1, 1].Source: §A.2 · p. 54
  • An alternate sixth-order solution, γ1 ≈ 1.6206645011137183, is A(α)-stable with α = 89.94077997° but has significantly larger relative period error.The study therefore does not pursue this member.Source: §A.2 · p. 54

A.3 Six sub-steps: s = 6

For six sub-steps, the method treats sub-step sizes as design parameters and relates the first size to dissipation or accuracy conditions. The resulting sixth- and seventh-order configurations involve a stability trade-off: the preferred seventh-order choice is nearly unconditionally stable, but not strictly so.

A.3 Six sub-steps: s = 6

  • For sixth-order accuracy, γ1 is related to user-specified ρ∞, while γ2 through γ6 remain free parameters; the stable branch retains controllable high-frequency dissipation for ρ∞ ∈ [0, 1].Admissible γ1 values combine the high-frequency dissipation condition with unconditional stability.Source: §A.3 · p. 57
  • The seventh-order conditions determine sub-step sizes by requiring local truncation errors of order O(∆t^8).The resulting candidate γ1 values do not yield a strictly unconditionally stable method within the framework.Source: §A.3 · pp. 57–58
  • No strictly unconditionally stable seventh-order six-sub-step method exists within framework (4).The selected practical configuration is A(α)-stable, with a stability angle extremely close to 90°.Source: §A.3 · p. 58
  • γ1 ≈ 0.55670 with ρ∞ ≈ −0.72080 gives the preferred seventh-order configuration, which is A(α)-stable with α ≈ 89.99999995°.It lies close to the strict unconditionally stable boundary.Source: §A.3 · p. 58
  • The alternative γ1 ≈ 1.89513, ρ∞ ≈ 0.52105 candidate has α = 89.85071945° and a larger relative period error.It also has less favorable stability properties and is not pursued.Source: §A.3 · p. 58
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