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A numerical method for two-dimensional Boussinesq-Full Dispersion systems modelling internal wave propagation
Angel Durán
TL;DR
Two-dimensional internal-wave models need numerical approximation and analysis of their periodic initial-value problems. The paper studies well-posedness, Fourier-Galerkin spatial approximation, and Implicit Midpoint Rule time integration. Its error estimates depend on the parameter group, with 2D estimates requiring more regularity in some cases and differing in optimal norm.
Problem
The paper addresses numerical approximation and analysis of periodic initial-value problems for two-dimensional Boussinesq-Full Dispersion internal-wave systems.
Source: §1 · pp. 1, 3Method
It analyzes periodic well-posedness and a spectral Fourier-Galerkin spatial discretization, then integrates the semidiscrete system with the Implicit Midpoint Rule.
Source: §1 · pp. 1, 3, §3 · p. 7Results
For G1, smooth solutions yield spectral convergence in 2D with more regularity required than in 1D and optimal L2 estimates for both variables; G2 and G3 have different dimension- and norm-dependent outcomes.
Source: §1 · pp. 3–4Takeaways & Limitations
The error estimates depend on the B-FD parameter group and dimension, so the 1D results do not uniformly describe the 2D approximation.
Source: §1 · pp. 3–4For G3 with a < 0 and c = 0, the 2D estimate is suboptimal in L2 for the interfacial wave, despite optimal H1 velocity estimates.
Source: §1 · p. 4, §3.2.3 · p. 13
Abstract
from arXiv · showhide
The paper is concerned with the numerical approximation of some two-dimensional asymptotic models for the propagation of internal waves in a two-layer system in the form of pde's for the interfacial deviation and the velocity. After the study of well-posedness of the periodic initial-value problem, the convergence of a semidiscrete approximation based on a spectral Fourier-Galerkin method is analyzed. The resulting semidiscrete system is then integrated numerically in time with the implicit midpoint rule. Some properties of the fully discrete scheme are described and its performance is illustrated with some numerical experiments.
1. Introduction
The paper analyzes a spectral Fourier-Galerkin approximation for two-dimensional Boussinesq-Full Dispersion internal-wave models, alongside periodic-problem well-posedness and implicit-midpoint time integration. Its error estimates depend on the parameter group and dimension, with distinct regularity and norm outcomes for the three system groups.
1. Introduction
- The paper introduces and analyzes a numerical method for 2D internal-wave models, extending the analysis of a 1D scheme to B-FD systems.It uses a Fourier-Galerkin spatial discretization and the Implicit Midpoint Rule for time integration.Source: §1 · pp. 1, 3
- The model describes waves along the interface of two inviscid, incompressible, homogeneous fluid layers, with a rigid lid above and a flat rigid bottom.The upper-layer density is lower than the lower-layer density.Source: §1 · p. 1
- The analysis considers three weakly dispersive groups: BBM-BBM, generic B-FD, and systems with only one of a or c negative.The 2D study is motivated by nonlocal terms represented through Fourier methods and by prior 2D work on Boussinesq/Boussinesq systems.Source: §1 · p. 3
- For G1, smooth exact solutions yield spectral convergence in 2D with more initial regularity than in 1D, and the estimates are optimal in L2 for both variables.The 2D estimate is also described as optimal order for the interfacial deviation and velocity.Source: §1 · pp. 3–4
- For G3 with a < 0 and c = 0, 2D velocity estimates remain optimal in H1, but the interfacial-wave estimate is suboptimal in L2.In 1D, this case has the same type of estimate as G2.Source: §1 · p. 4
- The Implicit Midpoint Rule is selected for stability and geometric properties useful in long-term simulations, though its diagonally implicit fixed-point iteration can be slower.The paper notes that explicit time discretizations may be an alternative for BBM-BBM systems.Source: §1 · p. 4
- The periodic initial-value problem has local existence and uniqueness results under parameter-dependent assumptions, with different Sobolev regularity for the interface and velocity when c = 0.For b, d > 0, a ≤ 0, c < 0, initial data lie in Hs for all variables; for c = 0, the interface data lie in Hs−1 and velocity data in Hs.Source: §2 · p. 7
3. Spectral semidiscretization
The section constructs a Fourier-Galerkin semidiscretization and establishes its conservation and convergence properties, with error estimates varying across B-FD parameter groups.
3. Spectral semidiscretization
- The Fourier-Galerkin approximation evolves the interfacial deviation and velocity in a finite-dimensional space of trigonometric polynomials.The initial data are projected onto that space, and projection-error estimates support the convergence analysis.Source: §3 · pp. 7–8
3.1. Conservation properties.
- When b = d, the semidiscrete solution conserves H for as long as it exists.The paper notes that collocation implementation then gives the semidiscrete system a Hamiltonian structure with a discrete version of H as Hamiltonian.Source: §3.1 · p. 8
3.2. Error estimates.
- For b, d > 0 and a = c = 0, the semidiscrete error estimate is optimal in L2 for both the interfacial deviation and velocity.The result assumes solution regularity µ > 1 and holds for sufficiently large N.Source: §3.2.1 · pp. 9, 11
- For the generic case b, d > 0 and a, c < 0, the error estimate is optimal in H1 for the interfacial deviation and velocity.The result assumes solution regularity µ > 3/2 and sufficiently large N.Source: §3.2.2 · p. 13
3.2.3. Group (G3): Other B-FD systems.
- In group G3, a = 0, c < 0 retains the generic-case estimate, whereas a < 0, c = 0 gives an estimate that is not optimal for the interfacial deviation in L2.The second parameter choice changes the order of α(k), leading to a different estimate.Source: §3.2.3 · p. 13
4. Full discretization and numerical experiments
The periodic problem is fully discretized with Fourier collocation in space and the implicit midpoint rule in time. Numerical experiments report second-order temporal convergence and show solitary-wave propagation with errors that stabilize over longer times.
4.1. Full discretization.
- The spatial discretization uses Fourier collocation on a uniform grid to represent the semidiscrete solution at collocation points.The nodal representation uses N × N Fourier pseudospectral differentiation matrices.Source: §4.1 · p. 14
- The collocation and Galerkin methods are algebraically equivalent in the cited sense, supporting an expectation of similar convergence results.The expectation is stated for the collocation scheme relative to the semidiscrete convergence propositions.Source: §4.1 · p. 15
- The implicit midpoint rule advances the semidiscrete system and is selected for its stability and geometric properties in long-term simulations.The fully discrete implementation operates on two-dimensional discrete Fourier coefficients, mode by mode.Source: §4.1 · p. 15
- For sufficiently small time steps, the fixed-point map is contractive on a bounded ball, yielding a bounded midpoint solution.The argument assumes the preceding time-level approximation exists and is bounded.Source: §4.1 · pp. 15–16
4.2. Numerical experiments.
- The manufactured-solution experiment shows second-order convergence in time, with L2 and L∞ errors reported at T = 1.The experiment uses N = 64; spectral-order convergence in space is expected for the regular solution.Source: §4.2 · p. 16, Table 3 · p. 16, Table 4 · p. 17
- The second experiment initializes line solitary waves from numerically generated profiles on a sufficiently long periodic interval and runs to T = 40.The reported case uses γ = 0.8, c_s = 0.5, x_0 = −10, and L = 32 with N = 256.Source: §4.2 · p. 17
- The solitary-wave experiment shows a traveling ζ profile with permanent shape and constant speed, while amplitude errors stabilize after growth to about t = 10.The discrete Hamiltonian error is also reported to suggest stabilization over time.Source: §4.2 · pp. 17–18, Figure 1 · p. 18, Figure 2 · p. 19