A high-order numerical model for nonlinear water waves remained stable for at least 50 wave periods in most of its long runs, but one demanding case exploded at about 49.5 periods. A separate test of a highly nonlinear wave hitting a vertical wall also broke down.
In a broader check of stabilization settings, 908 of 1,620 runs exploded, 225 were judged too damped and 487 finished. The authors called the success rate about 40%, while expecting runs with a zero hyperviscosity scale coefficient to explode.
What the model changes
The proposed method is an unfitted, high-order spectral-element framework for fully nonlinear potential-flow water waves. In an unfitted approach, the numerical mesh does not have to be made to conform exactly to the boundary. The calculation uses a polynomial-corrected shifted-boundary approximation.
The framework models fluid motion in two dimensions and assumes the fluid is incompressible, inviscid and irrotational. In ordinary terms, the model leaves viscosity and rotational flow out of its physical assumptions. At the free surface, the nonlinear conditions were imposed strongly by collocation, using finite differences of arbitrary order and an added numerical term called hyperviscosity. The method also uses polynomial-preserving gradient recovery to capture the vertical free-surface velocity.
The numerical evaluation used periodic and finite wave tanks. It covered mesh-refinement convergence, runtime, long propagation, wave generation and absorption, harmonic generation, and soliton interactions with walls and bathymetry changes.
Speed, then stability
One comparison produced a large gap in both error and runtime. A 2-by-2 mesh with polynomial order 6 and polynomial-preserving recovery had 3.03% error after 100 wave periods and took roughly 83 seconds. A 102-by-102 mesh with polynomial order 2 and local gradient recovery had 9.70% error and took around 13,319 seconds. The paper reports a 1,593-fold speed-up for the first comparison.
The authors explicitly describe the speed-up result as a small numerical study rather than a general proof. The wider stabilization sweep provides that context: 908 of 1,620 runs exploded, 225 were too damped and 487 completed. The reported success rate was about 40%, with runs that set the hyperviscosity scale coefficient to zero expected to explode.
Benchmark behavior
In a constant-depth generation and absorption test, profiles at 100 and 200 wave periods differed minimally, if at all. The authors interpreted that near-match as proper wave absorption.
Over a submerged bar, the wave steepened, and higher-order harmonics were generated and released. The analysis was reported to align visually with the cited experimental results.
In the vertical-wall soliton test, maximum run-up, attachment and detachment heights, and maximum horizontal force showed good agreement with digitized numerical reference results.
The computed results for a semi-circular bathymetry bump were described as agreeing qualitatively well.
The benchmark evidence was favorable, but not uniformly quantitative: the submerged-bar comparison was visual, the semi-circular-bump result was qualitative, and the wall comparison used digitized reference results.
The hardest cases
The wall test also exposed a hard limit. At the most nonlinear setting, with the initial soliton amplitude divided by water depth equal to 0.7, the calculation predicted only the attachment height, the initial force-profile segment and the maximum force. The text reports a breakdown when the free-surface profile became non-single-valued.
The manuscript is an arXiv preprint, version one, posted on 28 August 2026.
Paper data and sources
Original title: A high-order polynomial-corrected shifted boundary method for simulating fully nonlinear water waves
Authors: Jens Visbech, Allan P. Engsig-Karup, Harry B. Bingham, Mario Ricchiuto
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text