Leapfrog Crank--Nicolson decoupling of wave--heat-type problems

1. Introduction

• The framework is motivated by a thermo-elastic string, visco-elasticity with memory, thermo-elastic waves with finite thermal-signal speed, and Maxwell equations coupled to a dispersive interface material.Source: §1 · pp. 1–2
• Leapfrog advances the wave part while a Crank–Nicolson heat step evaluates the wave-coupling source at the temporal midpoint, keeping coupling explicit in a Strang-splitting-type scheme.The proposed scheme is intended to use the weaker hyperbolic CFL condition rather than the heat equation's usual parabolic time-step restriction.Source: §1 · p. 2
• The abstract spatial framework covers conforming and stabilized non-conforming wave discretizations, with stability under the desired CFL condition τ ≲ h.The analysis uses an error decomposition and summation-by-parts to avoid unphysical constraints on right-hand sides.Source: §1 · p. 2
• Prior decoupling work is described as scarce: analyzes stability for two second-order methods only for ordinary differential equations and gives no error bounds.Source: §1 · p. 3

Notation.

• The notation α ≲ β means α is bounded by a constant independent of the spatial mesh size h and time step τ.Source: Notation · p. 3

2. Analytical framework and main examples

• The weak framework organizes the wave and heat variables through bilinear forms and accommodates coupling expressed indirectly through boundary or interface terms.The paper notes that its assumptions also allow dissipative wave-type problems.Source: §2 · pp. 3–4
• The analysis could allow locally Lipschitz nonlinear contributions f or g, but omits them because they do not essentially change the analysis.Source: §2 · p. 5

2.1. Thermo-elastic string.

• For the thermo-elastic string, the framework uses the unusual state definition x = (∂tu, u), placing velocity before displacement.Source: §2.1 · p. 5

2.2. Non-local in time visco-elasticity and acoustics.

• The visco-elastic example treats elastic waves with non-local-in-time material laws, while a scalar visco-acoustic analogue replaces stress tensors with pressures.Source: §2.2 · pp. 5–6

2.3. Thermo-elastic wave coupling.

• The thermo-elastic wave example couples mechanical fields to heat conduction, representing the mechanical unknowns as x = (v, σ) and temperature as the heat variable.Source: §2.3 · p. 7

2.4. Maxwell equation with dissipative 2D material.

• The Maxwell example models a non-local-in-time graphene-like interface response using auxiliary interface fields coupled to bulk electric and magnetic fields.The interface coupling is included through the operator domain; the paper verifies the framework condition and well-posedness for this example.Source: §2.4 · pp. 8–9

• The framework permits nonconforming wave spaces and uses discrete bilinear forms for the wave and coupling terms.The spatial formulation also uses weaker-norm subspaces induced by p and ep.Source: §3 · p. 9
• The authors note that nonconforming heat discretizations, including mass lumping and symmetric interior penalty dG, could be treated but omit them for readability.Source: §3 · p. 9
• The discrete wave form preserves skew-symmetry up to a positive stabilization term, which acts only on discrete functions.Source: §3 · p. 9
• A consistency condition incorporates domain coupling information into the error analysis through integration by parts.Source: §3 · p. 10
• The error decomposition separates best-approximation errors, differential and coupling operator approximation, and right-hand-side approximation.The full error is split into discrete and best-approximation errors.Source: §3 · p. 10
• Theorem 3.4 bounds the discrete errors under Assumptions 2.1, 3.2, and 3.3, with constants independent of h.Source: §3 · p. 11

4.1. Derivation of the method.

• The scheme couples staggered leapfrog steps for the wave equation with a Crank–Nicolson heat step, replacing the coupling average with a computed half-step to avoid an implicit coupling.The half-step is described as a second-order approximation to the average; stabilization terms are applied so the wave steps remain explicit.Source: §4.1 · pp. 12–13

4.2. Stability.

• Under Assumption 4.4's CFL condition, the paper proves that the time integration scheme is stable.The stability proof uses energy estimates and a Gronwall-type argument for a perturbed evolution equation.Source: §4.2 · pp. 13, 15, 17

4.3. Error analysis.

• The error analysis derives an error recursion using defects as right-hand sides, estimates the Crank–Nicolson defect, and factorizes the LFCN defect.The analysis then applies the stability estimate to the error equation under the CFL condition.Source: §4.3 · pp. 17, 18, 19, 21
• Theorem 4.10 states that, for Nτ ≤ T, fully discrete errors satisfy its estimates under the stated regularity and Assumptions 2.1, 3.2, 3.3, and 4.4; constants depend on T and η, but not on h and τ.Source: §4.3 · pp. 20–21
• The estimates for I1 and I5 use Cauchy–Schwarz and Young’s inequality, while I4 is treated by summation by parts.The summation by parts is possible because the indicated operator does not depend on the time index.Source: §4.3 · pp. 22–23
• Cauchy–Schwarz and Young’s inequality bound the summands and boundary terms; stabilization terms and the boundary term at j = m are then absorbed.The constant for the j = m boundary term is chosen so it can be absorbed on the left-hand side.Source: §4.3 · pp. 22–23
• Setting m = m* in the preceding inequality and dropping the nonnegative stabilization term yields the stated bound.The preceding inequality also covers the alternative case mentioned immediately before this step.Source: §4.3 · p. 23
• Setting m = N, using the energy bound, and inserting the bounds from Lemmas 4.8 and 4.9 proves the claim.The inserted bounds concern Rj and ∂τ ˙x⋆,j.Source: §4.3 · pp. 23–24

5. Application to examples and numerical experiments

• The examples use shape-regular, quasi-uniform meshes, fixed polynomial degrees and material parameters, and satisfy the CFL condition in Assumption 4.4.Discrete derivatives are bounded by the maximum norms of their continuous counterparts without an additional constant.Source: §5 · p. 24

5.1. Thermo-elastic string.

• The thermo-elastic string has optimal-order spatial and fully discrete error bounds with constants independent of h and τ; the full bound holds under the CFL condition.For k≥2, the estimates suggest choosing heat degree ℓ=k−1.Source: §5.1 · p. 27

5.2. Non-local in time visco-elasticity and acoustics.

• For visco-elasticity, the analysis establishes optimal-order spatial and fully discrete error bounds; typical degree choices are ℓ=k or k+1.The section treats the visco-elastic case; acoustics is described as requiring only minor modifications.Source: §5.2 · pp. 27, 29

5.3. Thermo-elastic wave coupling.

• Thermo-elastic wave coupling has optimal-order spatial and fully discrete error bounds, with the latter valid under the CFL condition.The natural heat degree choice is ℓ=k.Source: §5.3 · p. 31

5.4. Maxwell equation with dissipative 2D material.

• For the Maxwell equation with dissipative 2D material, the paper derives spatial and fully discrete error bounds with constants independent of h and τ under the CFL condition.The analysis assumes a sufficiently regular solution and targets spatial rate r_h=h^k.Source: §5.4 · p. 32

6.1. Thermo-elastic string.

• The thermo-elastic string tests show second-order time convergence and stability for time steps below a threshold approximately proportional to h.The experiments also report optimal spatial convergence when the heat-space degree is one above the wave-space degree.Source: §6.1 · pp. 33–34, Figure 1 · p. 33, Figure 2 · p. 34
• For the thermo-elastic string, choosing heat degree k = ℓ + 1 gives the reported optimal spatial convergence.The exact solution is polynomial in time, so the experiment isolates spatial discretization error.Source: §6.1 · p. 34, Figure 3 · p. 35

6.2. Thermo-elastic wave coupling.

• In thermo-elastic wave coupling, the scheme exhibits second-order time convergence when the CFL condition τ ≲ h is satisfied.Second-order spatial ansatz functions resolve the spatial profile exactly in this temporal-convergence test.Source: §6.2 · p. 35, Figure 4 · p. 36
• For thermo-elastic wave coupling, equal wave and heat degrees yield order h^(p+1), while choosing ℓ = p − 1 yields h^p after the coarse-mesh pre-asymptotic regime.The equal-degree rate differs from the theoretical h^(p+1/2) estimate; the passage identifies this as a known upwind-flux stabilized dG phenomenon.Source: §6.2 · p. 36, Figure 5 · p. 37

6.3. Maxwell equation with dissipative 2D material.

• The discontinuous-profile Maxwell example exhibits the reported correct spatial convergence rate across three polynomial degrees.The experiment uses 20 target mesh sizes and a fixed time step of 0.0001.Source: §6.3 · p. 38, Figure 6 · p. 38
• The polynomial-profile Maxwell example confirms second-order time convergence and reports a linear CFL relation as the mesh is refined.The passage states that finer meshes impose a more severe CFL condition.Source: §6.3 · p. 38

A.1. Elastic example.

• The elastic discretization derives global discontinuous Galerkin forms from local forms and proves an auxiliary result needed for the analysis.The presentation follows the local discretization in.Source: §A.1 · p. 41
• The construction uses broken polynomial spaces on a matching triangulation, with piecewise-constant problem parameters and a fixed global ordering of mesh faces.The elastic setting is restricted to two dimensions and an isotropic elasticity tensor.Source: §A.1 · p. 42
• A global face ordering defines the normal and tangential vectors by the orientation on K1, with both vectors changing sign across the shared face.Local and global face conventions provide the basis for defining jumps and averages.Source: §A.1 · p. 42
• The operators Ss_h satisfy Assumption 3.2, as established in Lemma A.1.The proof uses the broken L2-orthogonal projection and approximation properties.Source: §A.1 · p. 44

A.2. Maxwell’s equations with interface coupling.

• For the Maxwell problem of Section 2.4, the analysis verifies Assumption 2.1 and proves the corresponding evolution equation is well-posed, while allowing piecewise-constant material parameters.The situation of Section 2.4 is recovered for ε = µ = 1.Source: §A.2 · p. 46
• The material-weighted inner product induces a norm equivalent to the L2(Ω)^6 norm when ε and µ are bounded above and below by positive constants.This establishes part (a) of Assumption 2.1.Source: §A.2 · p. 46
• Hodge projections extend the integration-by-parts identity from the divergence-free setting of to all test functions required by Assumption 2.1(c).The same projections allow sources J in L2(Ω)^3 that do not respect Gauss’s law.Source: §A.2 · p. 47
• The extended identity relates the volume curl terms to the tangential field and its interface jump.The formula is stated for the full curl spaces, without a divergence constraint.Source: §A.2 · p. 47

A.2.2. Well-posedness.

• The Maxwell system is written as an evolution equation with source terms and an operator domain that includes the interface condition.The formulation imposes no divergence constraints, allowing sources J in L2(Ω)^3.Source: §A.2.2 · p. 48
• The operator is accretive, and I + M is bijective, establishing maximal accretivity.Its energy estimate is nonnegative when the symmetric part of L is positive semidefinite.Source: §A.2.2 · pp. 48–49
• For the stated material and coupling assumptions, −M generates a contraction C0-semigroup and the evolution problem has a unique classical solution.The assumptions include positive-semidefinite symmetric part of L and nonzero cℓ.Source: §A.2.2 · p. 49