Abstract
We study diagonally implicit Runge–Kutta (DIRK) schemes when applied to abstract evolution problems that fit into the Gelfand-triple framework. We introduce novel stability notions that are well-suited to this setting and provide simple, necessary and sufficient, conditions to verify that a DIRK scheme is stable in our sense and in Bochner-type norms. We use several popular DIRK schemes in order to illustrate cases that satisfy the required structural stability properties and cases that do not. In addition, under some mild structural conditions on the problem we can guarantee compactness of families of discrete solutions with respect to time discretization.
Acknowledgement
The work of AJS is partially supported by NSF grant DMS-2111228. The work of IT was partially supported by LDRD-express project No. 223796 and LDRD-CIS project No. 226834 from Sandia National Laboratories, and by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics Program and by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research and Office of Fusion Energy Sciences, Scientific Discovery through Advanced Computing (SciDAC) program. IT wants to thank John N. Shadid for his continuous support and encouragement.
References
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A Example problems accommodating our assumptions
Let us present several examples of problems that our framework can manage. We will indicate when such problems satisfy our minimal set of assumptions, i.e., (2.2), and when they do satisfy the more stringent assumptions of Section 6. For a more comprehensive list and insight we refer the reader to [50, Ch. 8]. In all of the descriptions below d ≥ 1, and Ω ⊂ ℝd is a bounded domain with, at least, Lipschitz boundary.
A.1 Nonlinear diffusion equations
Let K : Ω × ℝ → ℝd×d be bounded, measurable, and nonnegative definite, i.e.,
The problem
can be cast into (1.1) by setting
Clearly this operator satisfies (2.2).
If, in addition, we assume that K is uniformly bounded, and uniformly positive definite, that is, there is K0 > 0 such that
then with
A.2 Nonlinear diffusion reaction problems
The previous example can be slightly generalized to
where γ : Ω × ℝ → ℝ is nonnegative for a nonnegative second argument, and it has sufficiently mild growth. For instance, if γ = |w|2w, we see that
so we get positivity, i.e., (2.2). If, in addition, d = 2 we recall that for
where we used Poincaré inequality. Now, the Gagliardo–Nirenberg interpolation inequality [50, Th. 1.24] implies that
Consequently, our problem fits into the framework of Section 6 with
A.3 Parabolic quasilinear equations
One further generalization that nonlinear diffusions allow is the following. Let G : Ω × ℝd → ℝ be convex in its second argument and F = D2G its derivative with respect to its second argument. Assume that these functions satisfy classical conditions of the form
with p > max{1, 2d/(d + 2)}. The equation
supplemented with suitable initial and boundary conditions, can be cast into the framework of Section 6 with p = q, ℍ = L2(Ω), and
A classical example of this scenario is the parabolic p–Laplacian problem
To see this, it suffices to set
A.4 The Navier–Stokes equations
The Navier–Stokes equations read
and are supplemented with suitable initial and boundary conditions. Here ν > 0 is the viscosity. To see how this problem fits the framework of Section 6 we set, for definiteness, d = 3 and
The operator 𝓐 is defined as
Owing to the skew symmetry of the convective term (over divergence free fields) we have
so that, clearly, C1 = ν and p = 2.
Consider now
where we, again, used the Gagliardo–Nirenberg interpolation inequality. This shows that
A.5 Hamiltonian problems
The operator 𝓐 is linear and induces a skew symmetric bilinear form on 𝕍, i.e.,
This is the prototypical case of Hamiltonian problems such as Maxwell’s equations in free space.
A.6 GENERIC systems
The operator 𝓐 is a combination of the cases in Sections A.1 and A.5, that is, a combination of a dissipative and a Hamiltonian parts. For instance we could consider, for ϵ ≥ 0, an operator of the form 𝓐(w) = 𝓢(w) + ϵ 𝓓(w) where, for all v, w ∈ 𝕍, we have
This type of PDE problems are usually called GENERIC [18, 21, 45, 46]. For instance, the linear wave equation with damping is a GENERIC system. Similarly, incompressible Navier–Stokes equations could be understood as the sum of a dissipative system (i.e., the bilinear form associated to viscous effects) and a nonlinear Hamiltonian system (the skew symmetric trilinear form associated to convective terms) (see, for instance, [44, 58]).
B Some properties of non-U-stable schemes
We start with a rather trivial observation.
Remark B.1
(non-U-stable schemes and skew-symmetric problems). Consider (1.1) with f ≡ 0 and 𝓐 a skew-symmetric operator, i.e.,
In other words, we consider purely autonomous dynamics. A non-U-stable two-stage DIRK scheme, meaning a scheme that does not satisfy the properties described in Definition 4.1, will satisfy the following discrete energy-balance
where the quadratic form 𝓠, introduced in (4.11), is unsigned. Similarly, a non-U-stable three-stage DIRK scheme, meaning a scheme that does not satisfy the properties described in Definition 5.1, will satisfy the discrete energy-balance
where 𝓠, defined in (5.11), is unsigned.
Identities (B.1) and (B.2) are an immediate consequence of (4.10) and (5.9) respectively. They follow from the fact that if 𝓐 is skew-symmetric, then we have that 〈𝓐(vn,i), vn,i〉 = 0 for each stage i ∈ {1, …, s}. Remark B.1 implies that non-U-stable schemes cannot be guaranteed to be stable when applied to problems of skew-symmetric nature. The same holds true for GENERIC-like PDEs (see Section A.6), with ϵ > 0 sufficiently small.
While simple, we think that this observation cannot be ignored or undermined. Linear as well as nonlinear autonomous problems, say ∂tu + 𝓐(u) = 0, can be decomposed into a symmetric and skew-symmetric parts, meaning
The symmetric part is usually positive and related to dissipative behavior. The skew-symmetric part describes either transport, conserved quantities, and/or wave-like nature of the problem. In this regard, Remark B.1 shows that unconditional stability cannot be expected when using non-U-stable schemes for PDEs strongly dominated by their skew-symmetric part. In this very specific context of non-U-stable schemes and problems dominated by its skew-symmetric operator it is pointless to attempt to develop any theory regarding convergence or error estimates; or to engage in any discussion related to order-reduction; or to compare its performance to other schemes. This is because, to begin with, the scheme cannot be proven to be stable. For many problems, say for instance the linear acoustic wave equations or Maxwell’s equations of electrodynamics, Remark B.1 should be the final argument against the use of non-U-stable schemes.
We notice, in particular, that Alexander’s DIRK22 and DIRK33 L-stable schemes, described by tableaus (3.7) and (3.11), respectively, are not U-stable. This narrows down the utility of these two schemes to the numerical solution of mildly nonlinear dissipative evolutionary PDEs. For the sake of completeness we present an optimal proof of stability for Alexander’s DIRK22 scheme in the context of linear, self-adjoint, and positive operators.
Proposition B.1
(energy identity I). Consider (1.1) with f ≡ 0. Assume that the operator 𝓐 is linear; coercive/ positive-definite, i.e., (6.1) holds with p = 2; and symmetric, that is,
Identity (4.8) for Alexander’s DIRK22 scheme takes the following specific form
with γ as defined in (3.7).
Proof
One only needs to use the values of the tableau, which are given in (3.7).□
As usual the problem lies with the unsigned off-diagonal term 〈𝓐 (vn,1), un+1〉 in the right-hand side of (B.3). We may consider absorbing part of it into the artificial damping terms as described in the following result.
Proposition B.2
(partial damping). Let ϰ ∈ ℝ be any real number. In the setting of Proposition B.1, the energy balance (B.3) can be rewritten as
where 𝓠ϰ(un, vn,1, un+1) is a quadratic form, depending on the free parameter ϰ, defined as
Proof
One needs to use techniques similar to those advanced in the proof of Lemma (4.2).□
Given the structure of (B.4)–(B.5) we may want to determine what is the optimal value of ϰ in order to preserve stability, at the very least when 𝓐 is a linear, positive-definite, symmetric operator:
Finding the optimal value of ϰ has two primary restrictions: we need 𝓠ϰ(un, vn,1, un+1) to remain non-negative; in addition, we also need to satisfy the property γ + ϰ > 0 in order to preserve the a priori bound on 〈𝓐 (vn,1), vn,1〉.
Setting ϰ = 1 – 2γ allows us absorb the off-diagonal term τn (1 – 2γ – ϰ) 〈𝓐 (vn,1), un+1〉, in its entirety, into the quadratic form 𝓠ϰ(un, vn,1, un+1). However, we already know from Section 4.2.3 that is not feasible. Since Alexander’s DIRK22 scheme is not U-stable, the choice ϰ = 1 – 2γ will lead to 𝓠ϰ(un, vn,1, un+1) being unsigned.
Some inspection reveals that the largest value of ϰ we can use, while also retaining non-negativity of 𝓠(un, vn,1, un+1) and positivity of γ + ϰ, is ϰ = γ.
These observations lead to the following result.
Lemma B.1
(a priori energy-estimate). Consider (1.1) with f ≡ 0. Assume that the operator 𝓐 is linear; coercive/positive-definite, i.e., (6.1) holds with p = 2; and symmetric, that is,
Then, the numerical solution using Alexander’s DIRK22 scheme satisfies the following optimal a priori energy estimate:
with 𝓠γ(un, vn,1, un+1) is defined in (B.5).
Proof
Estimate (B.6) is just a consequence of setting ϰ = γ in (B.4) and using Cauchy–Schwarz and Young’s inequality
for the unsigned off-diagonal term. We claim that (B.6) is optimal, in the sense that it maximizes the use of artificial damping terms while also preserving stability of the scheme.□
In conclusion, our assessment is that non-U-stable schemes may only be used either for positive linear problems, or positive nonlinear problems with very mild growth conditions. They may fail to be stable for problems strongly dominated by their skew-symmetric component. We assume that the arguments used in Propositions B.1 and B.2, and Lemma B.1, can be extended to the analysis of the Alexander’s DIRK33 scheme, but given the observations developed Remark B.1, we find very little motivations to do so.
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