arXiv subjects

Dimitrios Mitsotakis

Publications and source records attributed to Dimitrios Mitsotakis.

At least 19 recordsLinked to original sources

arXiv2026

A high-order regularization of the non-linear shallow water equations with weakly singular shock waves and its approximation by finite volume methods

Considered herein is a high-order regularization of the nonlinear shallow water equations within the framework of water wave theory. The regularized system is Galilean invariant and its solutions maintain an energy level that closely matches that of the nonlinear shallow water equations. However, in contrast to the classical nonlinear shallow water system, which admits discontinuous shock waves, the regularized formulation gives rise to weakly singular shock waves, which have continuous spatial profiles with unbounded spatial derivatives at isolated points. Using dynamical systems techniques, we establish the existence of such waves. Although weakly singular traveling waves remain continuous over their entire domain, their numerical approximation via finite element or pseudospectral schemes is affected by the emergence of spurious oscillations. To address this issue, we explore several finite volume methods for the accurate numerical approximation of these solutions. Our results demonstrate that the regularized system effectively reproduces the dynamics of the nonlinear shallow water equations in several scenarios. Moreover, our computations indicate that weakly singular shock waves are dynamically stable and can arise from general initial conditions connecting two asymptotic states. In contrast, other weakly singular structures, such as cusped solitons, appear to be structurally unstable, as we were unable to generate them from generic initial data.

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arXiv2025

A dichotomy of finite element spaces and its application to an energy-conservative scheme for the regularized long wave equation

Certain energy-conservative Galerkin discretizations for nonlinear dispersive wave equations have revealed an unusual convergence behavior: optimal convergence is attained when continuous Lagrange finite element spaces of odd polynomial degree are employed, whereas the use of even-degree polynomials leads to reduced accuracy. The present work demonstrates that this behavior is intrinsic to the structure of the finite element spaces themselves. In particular, it is shown to be closely connected to the standard $L^2$-projection of derivatives, which possesses a super-approximation property exclusively for odd polynomial degrees. We also examine the implications of this feature for an energy-conservative Galerkin approximation of the regularized long-wave equation where the energy is a cubic functional. Although the resulting scheme conserves both mass and energy, we further show that the impulse is approximated with high accuracy, and we establish {\em a priori} error bounds for the associated semi-discrete formulation.

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arXiv2025

Discrete Nonlinear Schr\"odinger versus Ablowitz-Ladik: Existence and dynamics of generalized focusing NLS-type lattices over a nonzero background

The question of well-posedness of the generalized focusing Ablowitz-Ladik and Discrete Nonlinear Schr\"{o}dinger equations with \textit{nonzero} boundary conditions on the infinite lattice is far less understood than in the case where the models are supplemented with vanishing boundary conditions. This question remains largely unexplored even in the standard case of cubic nonlinearities in which, in particular, the Ablowitz-Ladik equation is completely integrable while the Discrete Nonlinear Schr\"{o}dinger equation is not (in contrast with its continuous counterpart). We establish local well-posedness for both of these generalized nonlinear systems supplemented with a broad class of nonzero boundary conditions and, in addition, derive analytical upper bounds for the minimal guaranteed lifespan of their solutions. These bounds depend explicitly on the norm of the initial data, the background, and the nonlinearity exponents. In particular, they suggest the possibility of finite-time collapse (blow-up) of solutions. Furthermore, by comparing models with different nonlinearity exponents, we prove estimates for the distance between their respective solutions (measured in suitable metrics), valid up to their common minimal guaranteed lifespan. Highly accurate numerical studies illustrate that solutions of the generalized Ablowitz-Ladik equation may collapse in finite time. Importantly, the numerically observed blow-up time is in excellent agreement with the theoretically predicted order of the minimal guaranteed lifespan. Furthermore, in the case of the Discrete Nonlinear Schr\"{o}dinger equation on a finite lattice we prove global existence of solutions; this is consistent with our numerical observations of the phenomenon of \textit{quasi-collapse}, manifested by narrow oscillatory spikes that nevertheless persist throughout time ... continued in pdf

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arXiv2024

Bona-Smith-type systems in bounded domains with slip-wall boundary conditions: Theoretical justification and a conservative numerical scheme

Considered herein is a class of Boussinesq systems of Bona-Smith type that describe water waves in bounded two-dimensional domains with slip-wall boundary conditions and variable bottom topography. Such boundary conditions are necessary in situations involving water waves in channels, ports, and generally in basins with solid boundaries. We prove that, given appropriate initial conditions, the corresponding initial-boundary value problems have unique solutions locally in time, which is a fundamental property of deterministic mathematical modeling. Moreover, we demonstrate that the systems under consideration adhere to three basic conservation laws for water waves: mass, vorticity, and energy conservation. The theoretical analysis of these specific Boussinesq systems leads to a conservative mixed finite element formulation. Using explicit, relaxation Runge-Kutta methods for the discretization in time, we devise a fully discrete scheme for the numerical solution of initial-boundary value problems with slip-wall conditions, preserving mass, vorticity, and energy. Finally, we present a series of challenging numerical experiments to assess the applicability of the new numerical model.

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arXiv2024

Oscillatory and regularized shock waves for a modified Serre-Green-Naghdi system

The Serre-Green-Naghdi equations of water wave theory have been widely employed to study undular bores. In this study, we introduce a modified Serre-Green-Naghdi system incorporating the effect of an artificial term that results in dispersive and dissipative dynamics. We show that, over sufficiently extended time intervals, effectively approximates the classical Serre-Green-Naghdi equations and admits dispersive-diffusive shock waves as traveling wave solutions. The traveling waves converge to the entropic shock wave solution of the shallow water equations when the dispersion and diffusion approach zero in a moderate dispersion regime. These findings contribute to an understanding of the formation of dispersive shock waves in the classical Serre-Green-Naghdi equations and the effects of diffusion in the generation and propagation of undular bores.

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arXiv2023

The complex-step Newton method and its convergence

Considered herein is a modified Newton method for the numerical solution of nonlinear equations where the Jacobian is approximated using a complex-step derivative approximation. We show that this method converges for sufficiently small complex-step values, which need not be infinitesimal. Notably, when the individual derivatives in the Jacobian matrix are approximated using the complex-step method, the convergence is linear and becomes quadratic as the complex-step approaches zero. However, when the Jacobian matrix is approximated by the nonlinear complex-step derivative approximation, the convergence rate remains quadratic for any appropriately small complex-step value, not just in the limit as it approaches zero. This claim is supported by numerical experiments. Additionally, we demonstrate the method's robust applicability in solving nonlinear systems arising from differential equations, where it is implemented as a Jacobian-free Newton-Krylov method.

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arXiv2023

Mechanical balance laws for two-dimensional Boussinesq systems

Most of the asymptotically derived Boussinesq systems of water wave theory for long waves of small amplitude fail to satisfy exact mechanical conservation laws for mass, momentum and energy. It is thus only fair to consider approximate conservation laws that hold in the context of these systems. Although such approximate mass, momentum and energy conservation laws can be derived, the question of a rigorous mathematical justification still remains unanswered. The aim of this paper is to justify the formally derived mechanical balance laws for weakly nonlinear and weakly dispersive water wave Boussinesq systems. In particular, two asymptotic expansions used for the formal and rigorous derivation of the Boussinesq systems and the same are employed for the derivation and rigorous justification of the balance laws. Numerical validation of the asymptotic orders of approximation is also presented.

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arXiv2023

Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. II. Dynamics

Considered here are two systems of equations modeling the two-way propagation of long-crested, long-wavelength internal waves along the interface of a two-layer system of fluids in the Benjamin-Ono and the Intermediate Long-Wave regime, respectively. These systems were previously shown to have solitary-wave solutions, decaying to zero algebraically for the Benjamin-Ono system, and exponentially in the Intermediate Long-Wave regime. Several methods to approximate solitary-wave profiles were introduced and analyzed by the authors in Part I of this project. A natural continuation of this previous work, pursued here, is to study the dynamics of the solitary-wave solutions of these systems. This will be done by computational means using a discretization of the periodic initial-value problem. The numerical method used here is a Fourier spectral method for the spatial approximation coupled with a fourth-order, explicit Runge-Kutta time stepping. The resulting, fully discrete scheme is used to study computationally the stability of the solitary waves under small and large perturbations, the collisions of solitary waves, the resolution of initial data into trains of solitary waves, and the formation of dispersive shock waves. Comparisons with related unidirectional models are also undertaken.

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arXiv2023

On iterative methods based on Sherman-Morrison-Woodbury splitting

We consider a new splitting based on the Sherman-Morrison-Woodbury formula, which is particularly effective with iterative methods for the numerical solution of large linear systems. These systems involve matrices that are perturbations of circulant or block circulant matrices, which commonly arise in the discretization of differential equations using finite element or finite difference methods. We prove the convergence of the new iteration without making any assumptions regarding the symmetry or diagonal-dominance of the matrix. To illustrate the efficacy of the new iteration we present various applications. These include extensions of the new iteration to block matrices that arise in certain saddle point problems as well as two-dimensional finite difference discretizations. The new method exhibits fast convergence in all of the test cases we used. It has minimal storage requirements, straightforward implementation and compatibility with nearly circulant matrices via the Fast Fourier Transform. For this reasons it can be a valuable tool for the solution of various finite element and finite difference discretizations of differential equations.

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arXiv2023

Equations for small amplitude shallow water waves over small bathymetric variations

A generalized version of the $abcd$-Boussinesq class of systems is derived to accommodate variable bottom topography in two-dimensional space. This extension allows for the conservation of suitable energy functionals in some cases and enables the description of water waves in closed basins with well-justified slip-wall boundary conditions. The derived systems possess a form that ensures their solutions adhere to important principles of physics and mathematics. By demonstrating their consistency with the Euler equations and estimating their approximation error, we establish the validity of these new systems. Their derivation is based on the assumption of small bathymetric variations. With practical applications in mind, we assess the effectiveness of some of these new systems through comparisons with standard benchmarks. The results indicate that the assumptions made during the derivation are not overly restrictive. The applications of the new systems encompass a wide range of scenarios, including the study of tsunamis, tidal waves, and waves in ports and lakes.

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arXiv2022

Dispersive shocks in diffusive-dispersive approximations of elasticity and quantum-hydrodynamics

The aim is to assess the combined effect of diffusion and dispersion on shocks in the moderate dispersion regime. For a diffusive dispersive approximation of the equations of one-dimensional elasticity (or p-system), we study convergence of traveling waves to shocks. The problem is recast as a Hamiltonian system with small friction, and an analysis of the length of oscillations yields convergence in the moderate dispersion regime $\eps, \delta \to 0$ with $\delta = o(\eps)$, under hypotheses that the limiting shock is admissible according to the Liu E-condition and is not a contact discontinuity at either end state. A similar convergence result is proved for traveling waves of the quantum hydrodynamic system with artificial viscosity as well as for a viscous Peregrine-Boussinesq system where traveling waves model undular bores, in all cases in the moderate dispersion regime.

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arXiv2022

Extended water wave systems of Boussinesq equations on a finite interval: Theory and numerical analysis

Considered here is a class of Boussinesq systems of Nwogu type. Such systems describe propagation of nonlinear and dispersive water waves of significant interest such as solitary and tsunami waves. The initial-boundary value problem on a finite interval for this family of systems is studied both theoretically and numerically. First, the linearization of a certain generalized Nwogu system is solved analytically via the unified transform of Fokas. The corresponding analysis reveals two types of admissible boundary conditions, thereby suggesting appropriate boundary conditions for the nonlinear Nwogu system on a finite interval. Then, well-posedness is established, both in the weak and in the classical sense, for a regularized Nwogu system in the context of an initial-boundary value problem that describes the dynamics of water waves in a basin with wall-boundary conditions. In addition, a new modified Galerkin method is suggested for the numerical discretization of this regularized system in time, and its convergence is proved along with optimal error estimates. Finally, numerical experiments illustrating the effect of the boundary conditions on the reflection of solitary waves by a vertical wall are also provided.

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arXiv2022

Oscillatory and regularized shock waves for a dissipative Peregrine-Boussinesq system

We consider a dissipative, dispersive system of Boussinesq type, describing wave phenomena in settings where dissipation has an effect. Examples include undular bores in rivers or oceans where dissipation due to turbulence is important for their description. We show that the model system admits traveling wave solutions known as diffusive-dispersive shock waves, and we categorize them into oscillatory and regularized shock waves depending on the relationship between dispersion and dissipation. Comparison of numerically computed solutions with laboratory data suggests that undular bores are accurately described in a wide range of phase speeds. Undular bores are often described using the original Peregrine system which, even if not possessing traveling waves tends to provide accurate approximations for appropriate time scales. To explain this phenomenon, we show that the error between the solutions of the dissipative versus the non-dissipative Peregrine systems are proportional to the dissipation times the observational time.

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arXiv2022

Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre-Green-Naghdi equations

We consider strongly-nonlinear and weakly-dispersive surface water waves governed by equations of Boussinesq type, known as the Serre-Green-Naghdi system; it describes future states of the free water surface and depth averaged horizontal velocity, given their initial state. The lack of knowledge of the velocity field as well as the initial states provided by measurements lead to an ill-posed problem that cannot be solved by traditional techniques. To this end, we employ physics-informed neural networks (PINNs) to generate solutions to such ill-posed problems using only data of the free surface elevation and depth of the water. PINNs can readily incorporate the physical laws and the observational data, thereby enabling inference of the physical quantities of interest. In the present study, both experimental and synthetic (generated by numerical methods) training data are used to train PINNs. Furthermore, multi-fidelity data are used to solve the inverse water wave problem by leveraging both high- and low-fidelity data sets. The applicability of the PINN methodology for the estimation of the impact of water waves onto solid obstacles is demonstrated after deriving the corresponding equations. The present methodology can be employed to efficiently design offshore structures such as oil platforms, wind turbines, etc. by solving the corresponding ill-posed inverse water waves problem.

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arXiv2022

A regularized shallow-water waves system with slip-wall boundary conditions in a basin: Theory and numerical analysis

The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche's method. Convergence of the numerical method is analyzed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

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arXiv2021

A conservative fully-discrete numerical method for the regularised shallow water wave equations

The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge--Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods.

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arXiv2020

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations conserve all linear invariants and one nonlinear invariant for each system. The spatial semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially explicit, using relaxation Runge-Kutta methods. We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests.

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arXiv2020

Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with appropriate predictor-corrector method to achieve higher resolution. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed, thus unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.

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