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Stiff Norm Estimates for Variable-Stepsize Peer Two-Step Methods

Jens Lang, Bernhard A. Schmitt

arXiv:2608.26913v1math.NA

TL;DR

Stiff variable-stepsize Peer methods need rigorous norm estimates that remain valid across changing coefficient matrices. The paper constructs such estimates using a grid-independent left eigenvector and develops methods achieving high stage and global order. It reports order s + 1 with s stages, constructs methods through order 5, and extends the framework toward stiff optimal-control boundary value problems.

  • Problem

    Variable stepsizes make Peer propagation matrices form changing families, while rigorous uniform norm estimates for stiff semi-linear problems are needed.

  • Method

    The paper analyzes variable-stepsize L(α)-stable Peer methods with a grid-independent left eigenvector, using common transformations and norm criteria for their propagation matrices.

  • Results

    Order s + 1 is achieved with s stages, and the paper constructs L(α)-stable Peer methods through order 5 with uniform stability estimates on variable grids.

  • Takeaways & Limitations

    The framework supports high-stage-order stiff integration and provides a foundation for applications to stiff optimal-control boundary value problems.

  • Takeaways & Limitations

    The common-transformation analysis requires a grid-independent dominant left eigenvector, a severe restriction on general Peer methods.

Abstract

from arXiv · show

Peer two-step methods are attractive for the numerical solution of stiff initial value problems because they combine favorable features of Runge-Kutta and multi-step methods, including high-order accuracy, strong stability properties, and the avoidance of order reduction. This paper develops rigorous norm estimates for the propagation operators of variable-stepsize diagonally-implicit $L(α)$-stable Peer methods applied to stiff semi-linear systems. The analysis relies on a class of Peer methods whose propagation matrices possess a grid-independent left eigenvector, enabling the construction of suitable matrix norms for entire coefficient families. We show that this class contains $L(α)$-stable methods of stage order $q=s$ and super-convergence of order $s+1$ with $s$ stages, exceeding the classical stage-order barrier $q\le 2$ of irreducible $s$-stage diagonally-implicit Runge-Kutta methods, and construct such methods up to order 5. Achieving high stage order is one of the principal advantages of Peer methods, especially for stiff differential equations where it helps mitigate order reduction. The resulting theory provides uniform stability bounds for stiff semi-linear problems on variable grids and establishes a foundation for further applications to optimal control and related boundary value problems. Comparative numerical results are presented for three classical stiff benchmark problems, including three novel Peer two-step methods of order 3, 4, 5, and four singly diagonally implicit Runge-Kutta methods with a first explicit step of stage order 2 and order 3, 4, 5, and 6.

1 Introduction

The paper addresses the lack of rigorous norm tools for variable-stepsize Peer methods, whose coefficient matrices vary with stepsize ratios. It exploits a grid-independent left eigenvector to analyze these matrix families and supports high-order stiff integration.

  • Peer two-step methods combine Runge-Kutta and multi-step advantages, including strong stability properties and avoidance of order reduction for very stiff problems.
  • Variable stepsizes make Peer coefficients depend on stepsize ratios, complicating construction of one fixed norm for the resulting matrix families.
  • The analysis restricts attention to methods whose relevant left eigenvector of the propagation matrix is independent of the grid.
  • Rigorous variable-stepsize stability requires uniformly bounded long products of propagation matrices, a problem related to the joint spectral radius.
  • The paper uses the simplified Peer formulation for pure initial-value problems, while the redundant form provides additional degrees of freedom for adjoint problems.
  • The paper derives norm estimates, constructs three novel Peer methods and starting methods, analyzes global error, and reports numerical experiments.

2 Norm estimates for the stability matrix

The paper develops norm estimates for Peer stability matrices on unbounded left-half-plane regions, where spectral-radius bounds alone cannot control products on variable grids. The approach uses transformations, quadratic-form criteria, and holomorphic extension to obtain common norm bounds.

  • The Peer test-equation propagation matrix is R_n(z) = (I − z K̄_n)^−1 B̄_n, with z = h_nλ.
  • Variable-grid stability requires one index-independent norm that uniformly bounds products of different propagation matrices and can show damping as z approaches infinity.
  • The analysis transforms coefficient matrices to separate the dominant eigenvalue one from a southeast block, using the same similarity transformation for every grid factor.
  • A grid-independent dominant left eigenvector is required for this common transformation, imposing a severe restriction on general Peer methods.
  • The spectral norm criterion is expressed through positivity of a Hermitian matrix Ω(ξ) along complex rays, with Γ's symmetric and antisymmetric parts entering the test.
  • The maximum principle extends a ray-based estimate to the corresponding complex sector, while formal LU decomposition can support rigorous verification for all ξ > 0.

3 Peer two-step methods for initial-value problems

This section constructs LSRK Peer methods with grid-independent eigenvectors, enabling variable-grid norm estimates, high stage order, and super-convergence. It presents stability results and examples through order five, while noting a four-stage stability trade-off.

  • Construction: LSRK methods use a grid-independent dominant left eigenvector, allowing one similarity transformation and fixed norm across stepsize ratios.For cs = 1, the transformed left eigenvector is e1^T.
  • Order conditions: The LSRK condition makes the last Peer stage a Runge-Kutta stage and supports super-convergence by canceling the leading error term.The cancellation condition is e_s^Tβ_r = 0, and long propagation products converge to a rank-one matrix.
  • Order conditions: Order s + 1 can be achieved with s stages, including on general grids when the coefficient matrix is stepsize-ratio dependent.The additional order condition extends earlier super-convergence results for L(α)-stable Peer methods.
  • Stability analysis: Theorem 3.1 gives a uniform sectorial norm bound when the transformed matrix block is contractive and the damping parameters satisfy the stated restrictions.The bound applies across admissible stepsize ratios in S and uses a norm constructed from the transformed coefficient family.
  • Numerical methods: IP2o3 satisfies a fixed-norm estimate for σn ∈ {10/11, 1, 1.1} with damping factor ν = 1/20 in the sector |ℑz| ≤ −2ℜz.The reported estimate is |||(I − zK̄n)^−1B̄n||| ≤ 1/|1 − z/20|.
  • Numerical methods: An L(α)-stable three-stage order-4 method has α = 64.59°, while some four-stage L-stable methods reach α = 90° but exhibit stability concerns.The four-stage candidates have a second eigenvalue near one and ||B̄||∞ > 30, potentially amplifying rounding errors.

4 The global error for stiff initial value problems

The section derives uniform weighted-norm stability and global-error estimates for Peer two-step methods applied to stiff semilinear problems on variable grids. Under smooth-grid and spectral assumptions, the analysis establishes order s convergence generally and order s+1 super-convergence under additional conditions.

  • Stability framework: The weighted vector norm combines a method-dependent weight W with the eigenvector basis X of the stiff matrix J.This construction extends the norm to stage vectors in dimension s·m and controls the nonlinear term through cond_2(W ⊗ X).
  • Stability framework: Uniform stability requires one constant weight matrix W for all stepsize ratios, together with positive-semidefiniteness of the matrix family Ω(ξ).The resulting theorem applies to ratios in a prescribed set S and stepsizes bounded by 1/(2L_Φ).
  • Stability framework: The global error bound is independent of the stiffness norm ∥J∥_2 when the eigenvector condition number cond_2(X) is moderate.For normal J, the condition number is moderate, and the estimate may even imply error decay when ϖ = 0.9νμ + 2L_Φ < 0.
  • Global convergence: General variable grids yield convergence of order s after summing local errors of order s+1.The estimate is expressed using local derivative bounds over each interval.
  • Super-convergence: Order s+1 is obtained on smooth grids when the super-convergence condition and damping of the subdominant propagation modes hold.Theorem 4.2 assumes y⋆ ∈ C^{s+2}[0,T], smoothly varying stepsize ratios, and a bound on the southeast block of the transformed propagation matrix.
  • Super-convergence: The modified-solution argument raises the local truncation error from O(h_n^{s+1}) to O(h_n^{s+2}), producing the improved global order.The construction solves for correction vectors through the subdominant block and bounds them using its damping factor.

5 Boundary value problems in optimal control

The section extends Peer-method analysis to adjoint equations arising in ODE-constrained optimal control and studies the resulting boundary value problem. It derives stability and error bounds for the Pulcherrima triplet while documenting restrictions imposed by adjoint order conditions.

  • Problem formulation: ODE-constrained optimal control introduces an adjoint differential equation for a Lagrange multiplier, yielding a two-point boundary value problem.The control can be formally eliminated under suitable assumptions, leaving coupled state and adjoint variables.
  • Adjoint Peer methods: Adjoint Peer steps require additional adjoint order conditions, and redundant formulations provide extra degrees of freedom for satisfying them.The redundant form uses coefficient matrices A, B_n, and a constant positive definite diagonal K.
  • Adjoint Peer methods: For general stepsize ratios, coefficient restrictions force the Pulcherrima triplet to have order q = r = s − 1.The restrictions arise because a matrix related to B(σ) must be independent of σ, limiting stepsize dependence.
  • Stability analysis: The Pulcherrima standard method is L(α)-stable with α = 61.69° and tan α ≈ 1.85.The stability analysis uses transformed coefficient matrices and a fixed weight construction for the variable-ratio family.
  • Global error estimate: Theorem 5.1 gives boundary-value error estimates for sufficiently small stepsizes, while AP4o33vgi converges with order 3 for both state and adjoint variables.The theorem directly establishes order q − 1; a modification of the earlier argument yields the stated super-convergence order.

6 Numerical experiments

The experiments compare novel Peer two-step methods with ESDIRK methods on three stiff benchmarks, using uniform and adaptive variable stepsizes. Peer methods generally preserve higher-order behavior better, with IP4o5 consistently strongest in the reported tests.

  • Experimental setup: The study compares Peer methods of orders 3–5 with ESDIRK methods of orders 3–6 on three stiff benchmark problems.The comparisons use uniform stepsizes and adaptive local error control, with computing time and error as efficiency measures.
  • 6.1 Stiff Prothero-Robinson problem: Peer methods retain observed orders close to design order on Prothero-Robinson, while ESDIRK methods show serious order reduction for h > 10^-3.IP3o4 and IP4o5 perform considerably better than all ESDIRK methods before round-off errors appear for h < 10^-3.
  • 6.2 Singularly perturbed van der Pol system: IP4o5 performs by far best across van der Pol stiffness levels, while ESDIRK convergence rates decrease to stage order two when ε ≪ h.Peer methods roughly lose one order for ε = 10^-4 but regain full order for ε = 10^-6.
  • 6.2 Singularly perturbed van der Pol system: Higher-order ESDIRK methods are less efficient than lower-order ones for van der Pol because of order reduction, whereas higher-order Peer methods perform superior at sharper tolerances.IP5o4 is especially effective for sharper tolerances, while IP2o3 is not competitive.
  • 6.3 Stiff semi-discretized Burgers problem: IP4o5 is by far the most efficient Burgers method, outperforming the others by three orders of magnitude in adaptive runs.IP3o4 offsets three LU decompositions instead of one by achieving a higher fourth-order.

7 Conclusion

The paper establishes uniform variable-grid stability estimates for a restricted class of L(α)-stable Peer methods and demonstrates their high-order construction and numerical competitiveness. It also extends the proof technique to stiff boundary value problems arising in optimal control.

  • 7 Conclusion: The paper establishes rigorous, uniform stability bounds for certain variable-stepsize L(α)-stable Peer two-step methods applied to stiff semi-linear systems.The framework relies on a special grid-independent eigenstructure.
  • 7 Conclusion: L(α)-stable Peer methods with s stages can achieve stage order q = s and order s + 1, surpassing the traditional stage-order barrier.The paper constructs novel methods up to order 5 and supports efficient starting and iteration procedures.
  • 7 Conclusion: The proof technique also applies to stiff boundary value problems in optimal control and yields norm estimates for the AP4o33vgi triplet.The conclusion identifies this as an additional application of the stability analysis.
  • 7 Conclusion: Numerical experiments confirm the practicality and competitiveness of IP3o4 and IP4o5, with IP4o5 clearly outperforming established ESDIRK methods.This conclusion is based on the reported stiff benchmark comparisons.

A Coefficients of the Peer methods

The coefficient appendix specifies the node vectors and stepsize-dependent matrices used by the new initial-value Peer methods. It also identifies the stepsize-ratio scaling and starting-step coefficients needed for implementation.

  • A Coefficients of the Peer methods: The new s-stage methods IP(s)o(s + 1) specify a node vector c and lower triangular coefficient matrix K(σ), with B(σ) computed from them.The appendix presents the coefficient representation for the novel initial-value Peer methods.
  • A Coefficients of the Peer methods: The Vandermonde matrix Vs is formed from powers of the node vector c and is used with the matrix exponential ˜Es = exp(˜Es).The passage identifies the matrix construction but does not provide the full displayed formula.
  • A Coefficients of the Peer methods: The matrices Kn and Bn depend on the stepsize ratio σn = hn/hn−1, which also defines the diagonal scaling matrix Sn.The scaling uses powers of σn through the stage dimension.
  • A Coefficients of the Peer methods: Starting-step coefficients A0 and the diagonals of ˜A0 are supplied for the iterative solution, while their sub-diagonals coincide.These data complete the implementation specification for the starting step.

A.3 Coefficients of IP4o5 ,

The section reports the diagonal entries of the coefficient matrix ˜A0 for IP4o5.

  • The largest reported diagonal coefficient is 9.990214081789681.
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