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Conservative Three-Layer Schemes for Kirchhoff-Type Equations

Felix Krumbiegel, Jemal Rogava, Andreas Rupp, Zurab Vashakidze

arXiv:2609.27707v1math.NAcs.NA

TL;DR

The paper addresses temporal discretization of a two-dimensional Kirchhoff-type equation whose solution-dependent coefficient reflects global deformation energy. It analyzes two symmetric three-layer schemes, showing discrete energy conservation for the homogeneous constant-coefficient problem and establishing bounds, convergence, and a one-step solver for the nonlinear scheme.

  • Problem

    The equation is nonlocal because its Laplacian coefficient depends on the solution’s global deformation energy, and the paper considers parameters that may vary with time.

    Source: §1.1 · p. 2

  • Method

    The paper analyzes locally linear and genuinely nonlinear symmetric three-layer schemes, using discrete energies to obtain bounds and studying a fixed-point iteration for the nonlinear equation at each time level.

    Source: §5.1 · pp. 15–16, §7 · p. 20

  • Results

    Both schemes preserve discrete energies for the homogeneous constant-coefficient problem; the nonlinear scheme also has local second-order convergence and geometrically convergent iteration for sufficiently small τ.

    Source: §7 · p. 20

  • Takeaways & Limitations

    Index-local a priori bounds combined with the one-step solver construct the trajectory stepwise on the local interval those bounds cover, including for time-dependent coefficients.

    Source: §7 · p. 20

    The paper gives no known whole-interval existence-and-uniqueness result up to blow-up, and exact energy conservation has no established counterpart for time-dependent α and β.

    Source: §7 · p. 20

Abstract

from arXiv · show

We study a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions whose coefficients are allowed to depend on time, and we construct conservative discretizations in time for the associated initial--boundary value problem. We consider two symmetric three-layer schemes of Crank--Nicolson type, a locally linear one and a genuinely nonlinear one, and we show that each of them preserves a discrete analogue of the total mechanical energy of the homogeneous problem with constant coefficients. For the nonlinear scheme we establish uniform apriori bounds on the discrete solution and on its discrete velocity by working directly with the discrete energies, without invoking a nonlinear discrete Gr\"onwall inequality, the constants still grow exponentially in the final time, and we prove local second-order convergence in time, both for the solution and for the central-difference approximation of its first time derivative. The nonlinear system arising at each time level is solved by a fixed-point iteration: given iterates at the two preceding levels that satisfy the apriori bounds, that equation has exactly one solution and the iteration converges to it at a geometric rate once the time step is small enough. Since the apriori bounds are index-local, alternating them with that one-step solver constructs the trajectory stepwise, for time-dependent coefficients as well, on the local interval on which those bounds hold. Numerical experiments, carried out in a setting in which the spatial discretization contributes no error, exhibit the conservation of the discrete invariants, confirm the second order in time and verify the geometric convergence of the fixed-point iteration.

1 Introduction

The paper studies a two-dimensional Kirchhoff-type equation whose coefficient depends on global deformation energy, including when its parameters vary in time. It analyzes two three-layer time schemes and their energy conservation, bounds, convergence, and solution procedure.

1.1 Problem Formulation

  • The equation couples each point’s evolution to the membrane’s global deformation energy, while its parameters may vary in time.The model treats the solution-dependent coefficient as an effective tension; the problem is posed on a bounded convex domain with Lipschitz boundary.Source: §1.1 · p. 2

1.2 Related work and contribution

  • The closest cited prior work establishes convergence for a three-layer scheme for a one-dimensional string equation with time-dependent coefficients.Earlier work also considered Kirchhoff-type discretizations, including schemes for homogeneous problems with constant coefficients.Source: §1.2 · pp. 2–3
  • Two symmetric three-layer schemes, one locally linear and one genuinely nonlinear, preserve discrete analogues of mechanical energy for the homogeneous constant-coefficient problem.A separate quantity combining potential energy and a centered-difference kinetic-energy approximation converges to mechanical energy at second order as the temporal grid is refined.Source: §1.2 · p. 3
  • For the nonlinear scheme, direct discrete-energy estimates give uniform bounds and local second-order convergence for the solution and its central-difference time derivative.The estimates avoid a discrete Grönwall inequality, but their constants still depend exponentially on the final time.Source: §1.2 · p. 3
  • For sufficiently small time steps, the nonlinear equation at each level has a geometrically convergent iteration, which combines with index-local bounds to construct a trajectory stepwise.This construction also applies with time-dependent coefficients on the local interval covered by the bounds.Source: §1.2 · p. 3
  • A self-adjoint, positive-definite extension of the negative Laplacian broadens the framework to multiple dimensions and Lipschitz domains, with several boundary conditions when positivity is retained.The stated boundary-condition scope includes Dirichlet, Robin, mixed, and suitably restricted Neumann problems.Source: §1.2 · p. 3

2 Symmetric Three-Layer Semi-Discrete Schemes

The paper defines two symmetric three-layer semi-discrete schemes on a uniform temporal grid. Both are Crank–Nicolson type and second-order accurate, but one solves a linear problem at each layer while the other is nonlinear.

2 Symmetric Three-Layer Semi-Discrete Schemes

  • The schemes seek approximations on a uniform partition with time step τ, using grid-point forcing and discrete differences in the three-layer equation.The coefficients and nonlinearity are evaluated at the time level, and initial functions are specified to start the scheme.Source: §2 · pp. 3–4
  • The locally linear Crank–Nicolson-type scheme has second-order temporal accuracy and requires solving a linear problem at each layer.The stated accuracy is O(τ^2).Source: §2 · p. 4
  • The second scheme is also second-order accurate in time but is a genuinely nonlinear Crank–Nicolson-type scheme.It uses the same initial data as the preceding locally linear scheme.Source: §2 · p. 4

3 Energy Conservation for the Homogeneous Kirchhoff Equation with Constant Coefficients

For the homogeneous Kirchhoff equation with constant coefficients, the continuous mechanical energy is conserved. The paper derives discrete energy invariants for both semi-discrete schemes.

3 Energy Conservation for the Homogeneous Kirchhoff Equation with Constant Coefficients

  • The homogeneous constant-coefficient Kirchhoff equation conserves mechanical energy, including a nonlinear contribution to its potential energy.The conservation statement follows under the regularity conditions used in the energy calculation.Source: §3 · p. 4

3.1 An Analogue of the Energy Conservation Property for Semi-Discrete Numerical Schemes

  • The locally linear scheme preserves a discrete energy invariant across time levels.The derivation uses a relation among the discrete kinetic and potential-energy terms and a telescoping sum.Source: §3.1 · p. 5
  • The nonlinear scheme likewise preserves its discrete energy invariant at every time level.The paper identifies this invariant as replicating the continuous conservation property for the homogeneous problem.Source: §3.1 · p. 5

4 Investigation of the Crank–Nicolson-type Nonlinear Scheme

This section analyzes the nonlinear Crank–Nicolson-type scheme, using the self-adjoint positive Friedrichs extension of the spatial operator to establish uniform bounds and a regularity-dependent second-order convergence theorem for the solution and its central-difference time derivative. For the homogeneous problem with constant coefficients, the scheme’s discrete energy approximates the total energy to second order.

4.1 Principal Lemmas

  • The discrete velocity and L^1/2u_k are uniformly L2-bounded independently of the time step, without using a discrete Grönwall inequality.The bounds depend on the initial data, coefficients, final time, and source bound.Source: §4.1 · pp. 6–7
  • With additional operator-domain regularity, L^1/2 of the discrete velocity and Lu_k are locally uniformly bounded on a sufficiently short time interval.The interval may be shortened to ensure the estimate holds; the constants depend on its length.Source: §4.1 · pp. 7, 9

4.2 Convergence Analysis

  • The convergence theorem assumes a well-posed problem, operator-domain regularity, and additional temporal and spatial smoothness to obtain the scheme’s second-order time accuracy.Assumptions include a Lipschitz third time derivative and Lipschitz time-dependent mixed spatial derivatives in L2; the regularity also ensures the starting value satisfies the homogeneous Dirichlet condition.Source: §4.2 · p. 10
  • O(τ^2) in time is the local error order for the approximate solution.The estimate applies on a local time interval.Source: §4.2 · p. 13
  • O(τ^2) in time is also the local error order for the central-difference approximation of the solution’s first time derivative.The derivative is evaluated at discrete time levels.Source: §4.2 · p. 13
  • O(τ^2) is the difference between total energy and its discrete approximation for the homogeneous problem with constant coefficients.The discrete kinetic energy uses the central difference computed from the nonlinear scheme.Source: §4.2 · p. 14

5 Solvability of a Nonlinear Equation Arising from Temporal Discretization

The section proves existence and uniqueness for the nonlinear equation at one time level and geometric convergence of its fixed-point iteration under a small-step condition. It also explains how index-local bounds can be combined with the solver to construct a trajectory stepwise on the interval covered by those bounds.

5.1 Existence and Uniqueness of a Solution

  • Consecutive iterates contract for sufficiently small τ, and their limit satisfies the nonlinear equation; the same contraction condition yields uniqueness.The argument uses the positive definiteness of L to conclude equality of solutions.Source: §5.1 · pp. 15–16

5.2 Geometric Convergence of the Iterative Method

  • The error between an iterate and the solution converges at the same geometric rate as the contraction between consecutive iterates.This estimate is made only after one-step existence and uniqueness have been established.Source: §5.2 · pp. 16–17
  • For preceding iterates satisfying the stated bounds, the nonlinear equation has exactly one solution when 0 < τ < τ0, and iteration converges with factor ρ = τ/τ0 < 1.The threshold depends on problem data, not on the time level, number of steps, or τ.Source: §5.2 · p. 17
  • The one-step theorem alone does not establish the full trajectory, but index-local energy bounds can be alternated with the solver to construct it stepwise on the local interval covered by the bounds.The construction uses the data regularity required by Lemma 4.3.Source: §5.2 · p. 17

6 Numerical Experiments

The experiments test temporal discretization in a spatially exact setting, finding discrete-invariant conservation up to rounding error and second-order temporal convergence. They also show geometric convergence of the nonlinear iteration and second-order accuracy with time-dependent coefficients.

6 Numerical Experiments

  • A Laplacian eigenfunction and initial data in its span reduce the evolution exactly to a scalar three-layer recursion, eliminating spatial-discretization error while retaining the nonlinearity.The nonlinear scheme reduces to a scalar cubic equation, solved by Newton’s method to tolerance 10−15.Source: §6 · p. 17
  • Over 200,000 steps on [0, 100], both schemes’ discrete invariants remain constant to rounding error, and refinement increases the drift, indicating an identity rather than an approximation.The run uses τ = 5 · 10−4, α = β = 1, and λ = 1.Source: Figure 6.1 · p. 18, §6 · p. 18
  • Both schemes attain second-order convergence in time; on this test, the locally linear scheme has smaller error at every tested step size and requires no nonlinear solve.The solution error is measured at T = 10 against a high-accuracy reference solution.Source: §6 · pp. 18–19
  • The fixed-point iteration converges geometrically at every time level, while its observed contraction ratio falls by about a factor of four whenever τ is halved.The reported iteration count is the largest number needed at any time level to reduce the increment below 10−14.Source: §6 · p. 19, Table 6.1 · p. 19
  • For the nonautonomous test with α(t)=1+4t and β(t)=2−4t, Figure 6.3 reports relative error at the final time; both three-layer schemes attain the optimal second-order rate.The example lets the solution grow exponentially. The coefficients satisfy α≥1 and β≥1 on their respective intervals.Source: §6 · p. 19, Figure 6.3 · p. 20

7 Conclusion

The two symmetric three-layer schemes preserve their respective discrete energies, while the continuous energy is approximated rather than conserved. For the nonlinear scheme, energy-based bounds support local second-order convergence and a geometrically convergent one-step solver, but the results do not establish whole-interval existence and uniqueness or exact conservation for time-dependent coefficients.

7 Conclusion

  • Both schemes preserve their respective discrete energies identically at every time level; neither conserves the continuous energy E(t), which E*_k approximates to O(τ^2).The conservation result concerns the homogeneous problem with constant coefficients.Source: §7 · p. 20
  • For the nonlinear scheme, direct use of discrete energies gives uniform bounds on the discrete velocity and L^1/2 u_k, plus local bounds that yield second-order time convergence for the solution and its central-difference velocity.The bounds and convergence are local in time.Source: §7 · p. 20
  • At each time level, the nonlinear equation has a unique solution and its fixed-point iteration converges geometrically for sufficiently small τ; together with index-local bounds, this constructs the trajectory stepwise on the interval covered by those bounds.The conclusion states that this stepwise construction also applies with time-dependent coefficients.Source: §7 · p. 20
  • A result guaranteeing existence and uniqueness of a discrete solution over the whole interval [0,T] until blow-up is not known to the authors.Exact energy conservation is established for the homogeneous constant-coefficient problem; for time-dependent α and β, the authors expect controlled drift instead.Source: §7 · p. 20

Declarations

The authors report funding from German and Georgian research organizations, no further funding or competing interests, and data and figure sources in the accompanying repository.

Declarations

  • The work reports support from the DFG, including a project grant and Germany’s Excellence Strategy, and from Georgia’s SRNSFG.The authors declare no further funding.Source: Declarations · p. 20
  • The authors declare that they have no competing interests.Source: Declarations · p. 20
  • Data underlying Section 6 figures and tables, along with figure sources, are available in the accompanying repository.Source: Declarations · p. 20
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