Numerical methods for evolution equations are designed based on accuracy and stability. The theory of both is highly developed for constant time step and linear problems. Less is known for variable time steps and nonlinear problems.These cases are subtle. For example, for increasing time steps, the BDF2 method loses A-stability and suffers non-physical energy growth in the approximate solution . Even the trapezoidal method is unstable when used with variable time steps, see e.g. , [3, pp. 181-182]. Dahlquist, Liniger and Nevanlinna in  proposed a one parameter -family of variable-step, one-leg, two-step methods (DLN), which are second-order accurate, and variable-step, nonlinearly, long-time stable. Its detailed specification (given in Section 2), is sufficiently Gordian to deter its use in complex applications, in which a method with DLN’s excellent properties should be valued. Our preliminary work on adaptive time-stepping for flow problems [4, 5] shows that (DLN) has promise, motivating the work herein.
Refactorization generally means restructuring of an existing algorithm without changing its behaviour. The goal of refactorization is to reduce complexity by creating a simple and clean logical structure, improving implementation, code readability, source maintainability and extensibility. Herein we show how (DLN) can be refactorized to be easily implemented in an intricate, possibly legacy/black-box code, without modifying the ‘assemble and solve’ portion. While our refactorization can work for other base methods, to fix ideas for , we consider a method based on the fully implicit Euler method
Figure 1 illustrates the implementation of the (DLN) method in Algorithm 1, by adding a pre-filter step to the data ahead of the nonlinear solver (BE), and a poster-filter step after the solver (BE). This algorithmic idea is our first contribution. In Section 3 we prove a new expression for the local truncation error, which simplifies the time-adaptive implementation of (DLN), and also recall its variable step -stability property.
2 The DLN method and its refactorization
We consider a numerical approximation of the initial value problem
Note that are step size independent, while are step size dependent. Define the average step size as follows:
The variable step DLN method of  as a one-leg 111 The ‘one-leg’ term was coined by Dahlquist in 1975  to name the multistep methods which involve only one value of in each step. In particular, the leapfrog and BDF methods are one-leg multistep methods. method is
To reduce the complexity of implementing (DLN), we consider its implementation through
pre- and post-processes of an implicit (backward) Euler method,
described in Algorithm 1, and illustrated in
// Define the refactorization coefficients
, If desired: Estimate Error and adapt
Since the coefficients satisfy
2.1 Related Work
The (DLN) method is variable-step -stable outgrowth of a method of Liniger , which is non-autonomous -stable (i.e. for ). The pre- and post-process steps in the Algorithm 1 are akin to time filters, highly developed as numerical methods in atmospheric science [10, 11, 12, 13, 14]. Recently it was noticed in  that this technique for adding stability can also increase accuracy. The idea of prefilter simple method postfilter was develloped in a different direction for constant time steps in .
The refactorization of an algorithm to reduce its cognitive complexity has been used in  to rearrange a family of one-leg one-step methods into a backward Euler code followed by post-processing, and further applied for partitioning multi-physics problems [17, 18, 8, 19]. In , the authors describe the implementation of the (DLN) formulas in a Nordsieck formulation [21, 22] essentially identical to that of the backward differentiation formulas, facilitating to adapt Nordsieck formulation codes like DIFSUB [23, 24] to the (DLN) formulas.
3 Convergence analysis of (Dln)
3.1 Consistency error
In [25, 26], the variable time-step (DLN) method was implemented in an adaptive manner, using a local and global error estimator. Similar to , the authors of [20, 25, 26, 6] use the classical definition of the local truncation error
where . The above definition follows the approach taken in the analysis of linear multistep methods (see e.g. [27, page 27]), and involves both the differentiation defect , and the interpolation defect . Dahlquist raised in  the question of the appropriateness of this viewpoint: “We accept this definition, but we do not accept as the adequate local truncation error!”
Using the refactorized form (DLN2BE) and Theorem 2.1, we now prove that the local truncation error of the one-leg (DLN) method can be evaluated only by the differentiation defect (3.1), similarly to the midpoint rule  and the Runge-Kutta methods. The new expression (3.1) simplifies greatly the error estimation.
The local truncation error of (DLN) is the differentiation error and
The consistency order and the coefficient of the leading term in (3.1) follow by Taylor expansions. On one hand, since (DLN) can be refactorized as the one-step method (DLN2BE), we further write the (DLN) method as follows
On the other hand, when we integrate (2.1) on and on , multiply the results by and , respectively, and add, we obtain
Finally, approximating both integrals in the right hand-side with the chord quadrature rule, with as the point of evaluation on both intervals, gives
In particular, for and , from (3.1) we have that
Let and denote the inner product and -norm in Euclidean space . For any pair of solutions of (2.1), a necessary and sufficient condition [29, page 384] for to be a non-increasing function of is the contractivity (one-sided Lipschitz) condition on :
The system (2.1) for which satisfies the (contractivity) condition is called dissipative, see e.g. the Definition in [30, page 268]. We recall that a Runge-Kutta method is B-stable, if the (contractivity) condition implies for any numerical solutions, see e.g. [31, page 359], or Definition 12.2 in . Similarly, a 2-step linear multistep method is called -stable [33, 29, 34, 32] if there exists a real positive definite matrix such that its one-leg version is contractive, namely , where . In the case of the (DLN) method, there exists such a positive definite matrix (independent of the step size)
As pointed out by Dahlquist in , both -stability and -stability imply -stability, and -stability implies -stability for constant time steps.
The (DLN) method is unconditionally -stable, and
where the -coefficients are
The ‘energy’ identity (3.2), implicit in , follows from algebraic manipulations, see e.g. . The -stability of (DLN), i.e. , follows from (3.2) and the (contractivity) assumption. The only (DLN) methods which yield the invariance of the solution are the symplectic (one-step midpoint) and (two-step midpoint) rules: the numerical dissipation vanishes if and only if .
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