arXiv subjects

Mukul Dwivedi

Publications and source records attributed to Mukul Dwivedi.

12 recordsLinked to original sources

arXiv2026

A numerical method for the fractional Zakharov-Kuznetsov equation

This paper develops a fully discrete Fourier spectral Galerkin (FSG) method for the fractional Zakharov--Kuznetsov (fZK) equation posed on a two-dimensional periodic domain. The equation generalizes the classical ZK model by replacing the Laplacian with a fractional Laplacian of order $α\in(0,2]$, thereby covering the classical ZK equation $α=2$, the higher-dimensional Benjamin--Ono--ZK equation $α=1$, and weaker fractional-dispersion regimes $0<α<1$. We first propose a semi-discrete FSG scheme in space that preserves the discrete analogues of mass, momentum, and Hamiltonian energy. Using periodic Kato--Ponce product and commutator estimates, we prove local-in-time uniform Sobolev bounds and strong convergence of the semi-discrete approximations to the unique strong solution in $C([0,\bar T];L^2_{\mathrm{per}}(Ω))$, for the initial condition in $H^s_{\mathrm{per}}(Ω)$, $s\geq 2+α$, and, as by product, we show that the existence and uniqueness of fZK equation in $L^\infty(0,\bar T;H^s_{\mathrm{per}}(Ω))\cap W^{1,\infty}(0,\bar T;L^2_{\mathrm{per}}(Ω))$. We then introduce a modified projection adapted to the fractional transport dispersive operator and prove optimal spatial error estimates of order $\mathcal O(N^{-r})$ for $r>2+α$, together with exponential convergence for analytic solutions. An integrating-factor fourth-order four-stage Runge--Kutta time discretization is used to integrate the stiff fractional dispersive part exactly, and a fourth-order temporal error estimate is obtained under a high-regularity nonlinear stability assumption. Numerical experiments illustrate the accuracy, fractional-order dependence, and fully discrete conservation drift of the method.

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arXiv2026

A Convergent Hybridizable Discontinuous Galerkin Method for Einstein--Scalar Equations

We propose and analyze a hybridized discontinuous Galerkin (HDG) method for the spherically symmetric Einstein--scalar system in Bondi gauge. After rewriting the model as a local first-order PDE--ODE system by introducing suitable scaled variables, we construct a semidiscrete scheme in which the element unknowns are computed locally and the coupling is carried by traces on the mesh skeleton. In the present radial setting, these traces can be eliminated recursively, so that only the main evolution variable is advanced in time, while the metric variables are recovered from discrete constraint relations. We prove local semidiscrete well-posedness, derive a global $L^2$--stability estimate, establish an optimal order $L^2$ error bound for the main evolution variable for polynomial degree $k\ge 1$, and obtain reconstruction error estimates for the metric variables and the associated mass functional. Numerical experiments verify the predicted spatial convergence rate and illustrate qualitative features of the Einstein--scalar dynamics, including large-data collapse profiles and smooth-pulse evolution.

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arXiv2026

A Galerkin Finite Element Method for the Fractional Calderón Problem

We study a numerical reconstruction strategy for the potential in the fractional Calderón problem from a single partial exterior measurement. The forward model is the fractional Schrödinger equation in a bounded domain, with prescribed exterior Dirichlet datum and corresponding measurement of the exterior flux in an open observation set. Motivated by single-measurement uniqueness results based on unique continuation \cite{ghosh2020uniqueness}, we propose a decomposition strategy and a Galerkin--Tikhonov method to recover the potential by a stabilized least-squares quotient in a dedicated coefficient space. We prove the existence and uniqueness of the discrete reconstructor and establish conditional convergence under natural consistency and parameter choice assumptions. We further derive {\it a priori} error estimates for the reconstructed state and for the coefficient reconstruction, and combine the latter with logarithmic stability for the continuous inverse problem to obtain a total coefficient error bound. The framework cleanly separates the forward solver from the inverse reconstruction step and is compatible with practical truncation and quadrature schemes for the integral fractional Laplacian. Numerical experiments in one and two space dimensions illustrate stability with respect to noise and demonstrate reconstructions of both smooth and discontinuous potentials.

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arXiv2026

A hybridizable discontinuous Galerkin method for the Ostrovsky equation

This paper develops the hybridizable discontinuous Galerkin (HDG) method for the Ostrovsky equation, a nonlinear dispersive wave equation featuring both third-order dispersion and a nonlocal antiderivative term with Coriolis effect. On a bounded interval, the nonlocal operator $\partial_x^{-1}$ is localized through an auxiliary variable $v$ satisfying $v_x=u$ together with an additional boundary constraint that ensures uniqueness. We employ a mixed first-order formulation to decompose the dispersive operator and to localize the nonlocal term, and we couple the resulting semi-discrete HDG scheme with a $θ$-time stepping method for $θ\in [1/2,1]$. We prove $L^2$-stability for suitable stabilization parameters and derive an {\it a priori} $L^2(Ω)$ error estimate for smooth solutions that explicitly accounts for the nonlinear convective flux. Numerical examples illustrate the convergence properties and demonstrate the scheme's capability to handle smooth and non-smooth solutions, including solitary wave propagation and peaked solitary wave (peakon) propagation in the zero dispersive limit regime.

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arXiv2026

A Hybridizable Discontinuous Galerkin Method for the non--local Camassa--Holm--Kadomtsev--Petviashvili equation

This paper develops a hybridizable discontinuous Galerkin method for the two-dimensional Camassa--Holm--Kadomtsev--Petviashvili equation. The method employs Cartesian meshes with tensor-product polynomial spaces, enabling separate treatment of $x$ and $y$ derivatives. The non-local operator $\partial_{x}^{-1}u_{y}$ is localized through an auxiliary variable $v$ satisfying $v_x = u_y$, allowing efficient element-by-element computations. We prove energy stability of the semi-discrete scheme and derive $\mathcal{O}(h^{k+1/2})$ convergence in space. Numerical experiments validate the theoretical results and demonstrate the method's capability to accurately resolve smooth solutions and peaked solitary waves (peakons).

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arXiv2025

A convergent Fourier spectral Galerkin method for the fractional Camassa-Holm equation

We analyze a Fourier spectral Galerkin method for the fractional Camassa-Holm (fCH) equation involving a fractional Laplacian of exponent $α\in [1,2]$ with periodic boundary conditions. The semi-discrete scheme preserves both mass and energy invariants of the fCH equation. For the fractional Benjamin-Bona-Mahony reduction, we establish existence and uniqueness of semi-discrete solutions and prove strong convergence to the unique solution in $ C^1([0, T];H^α_{\mathrm{per}}(I))$ for given $T>0$. For the general fCH equation, we demonstrate spectral accuracy in spatial discretization with optimal error estimates $\mathcal{O}(N^{-r})$ for initial data $u_0 \in H^r(I)$ with $r \geq α+ 2$ and exponential convergence $\mathcal{O}(e^{-cN})$ for smooth solutions. Numerical experiments validate orbital stability of solitary waves achieving optimal convergence, confirming theoretical findings.

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arXiv2025

Spectral Galerkin method for the zero dispersion limit of the fractional Korteweg-de Vries equation

We present a fully discrete Crank-Nicolson Fourier-spectral-Galerkin (FSG) scheme for approximating solutions of the fractional Korteweg-de Vries (KdV) equation, which involves a fractional Laplacian with exponent $α\in [1,2]$ and a small dispersion coefficient of order $\varepsilon^2$. The solution in the limit as $\varepsilon \to 0$ is known as the zero dispersion limit. We demonstrate that the semi-discrete FSG scheme conserves the first three integral invariants, thereby structure preserving, and that the fully discrete FSG scheme is $L^2$-conservative, ensuring stability. Using a compactness argument, we constructively prove the convergence of the approximate solution to the unique solution of the fractional KdV equation in $C([0,T]; H_p^{1+α}(\mathbb{R}))$ for the periodic initial data in $H_p^{1+α}(\mathbb{R})$. The devised scheme achieves spectral accuracy for the initial data in $H_p^r,$ $r \geq 1+α$ and exponential accuracy for the analytic initial data. Additionally, we establish that the approximation of the zero dispersion limit obtained from the fully discrete FSG scheme converges to the solution of the Hopf equation in $L^2$ as $\varepsilon \to 0$, up to the gradient catastrophe time $t_c$. Beyond $t_c$, numerical investigations reveal that the approximation converges to the asymptotic solution, which is weakly described by the Whitham's averaged equation within the oscillatory zone for $α= 2$. Numerical results are provided to demonstrate the convergence of the scheme and to validate the theoretical findings.

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arXiv2025

Stability of fully Discrete Local Discontinuous Galerkin method for the generalized Benjamin-Ono equation

The main purpose of this paper is to design a fully discrete local discontinuous Galerkin (LDG) scheme for the generalized Benjamin-Ono equation. First, we proved the $L^2$-stability for the proposed semi-discrete LDG scheme and obtained a sub-optimal order of convergence for general nonlinear flux. We develop a fully discrete LDG scheme using the Crank-Nicolson (CN) method and fourth-order fourth-stage Runge-Kutta (RK) method in time. Adapting the methodology established for the semi-discrete scheme, we demonstrate the stability of the fully discrete CN-LDG scheme for general nonlinear flux. Additionally, we consider the fourth-order RK-LDG scheme for higher order convergence in time and prove that it is strongly stable under an appropriate time step constraint by establishing a \emph{three-step strong stability} estimate for linear flux. Numerical examples associated with soliton solutions are provided to validate the efficiency and optimal order of accuracy for both methods.

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arXiv2024

Analysis of a local discontinuous Galerkin scheme for fractional Korteweg-de Vries equation

We propose a local discontinuous Galerkin (LDG) method for the fractional Korteweg-de Vries (KdV) equation, involving the fractional Laplacian with exponent $α\in (1,2)$ in one and multiple space dimensions. By decomposing the fractional Laplacian into first-order derivatives and a fractional integral, we prove the $L^2$-stability of the semi-discrete LDG scheme incorporating suitable interface and boundary fluxes. We derive the optimal error estimate for linear flux and demonstrate an error estimate with an order of convergence $\mathcal{O}(h^{k+\frac{1}{2}})$ for general nonlinear flux utilizing the Gauss-Radau projections. Moreover, we extend the stability and error analysis to the multiple space dimensional case. Additionally, we discretize time using the Crank-Nicolson method to devise a fully discrete stable LDG scheme, and obtain a similar order error estimate as in the semi-discrete scheme. Numerical illustrations are provided to demonstrate the efficiency of the scheme, confirming an optimal order of convergence.

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arXiv2024

Fully discrete finite difference schemes for the Fractional Korteweg-de Vries equation

In this paper, we present and analyze fully discrete finite difference schemes designed for solving the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation involving the fractional Laplacian. We design the scheme by introducing the discrete fractional Laplacian operator which is consistent with the continuous operator, and posses certain properties which are instrumental for the convergence analysis. Assuming the initial data (u_0 \in H^{1+α}(\mathbb{R})), where (α\in [1,2)), our study establishes the convergence of the approximate solutions obtained by the fully discrete finite difference schemes to a classical solution of the fractional KdV equation. Theoretical results are validated through several numerical illustrations for various values of fractional exponent $α$. Furthermore, we demonstrate that the Crank-Nicolson finite difference scheme preserves the inherent conserved quantities along with the improved convergence rates.

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arXiv2023

Convergence of a conservative Crank-Nicolson finite difference scheme for the KdV equation with smooth and non-smooth initial data

In this paper, we study the stability and convergence of a fully discrete finite difference scheme for the initial value problem associated with the Korteweg-De Vries (KdV) equation. We employ the Crank-Nicolson method for temporal discretization and establish that the scheme is $L^2$-conservative. The convergence analysis reveals that utilizing inherent Kato's local smoothing effect, the proposed scheme converges to a classical solution for sufficiently regular initial data $u_0 \in H^{3}(\mathbb{R})$ and to a weak solution in $L^2(0,T;L^2_{\text{loc}}(\mathbb{R}))$ for non-smooth initial data $u_0 \in L^2(\mathbb{R})$. Optimal convergence rates in both time and space for the devised scheme are derived. The theoretical results are justified through several numerical illustrations.

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arXiv2023

Stability and Convergence analysis of a Crank-Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

In this paper we study the convergence of a fully discrete Crank-Nicolson Galerkin scheme for the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation, which involves the fractional Laplacian and non-linear convection terms. Our proof relies on the Kato type local smoothing effect to estimate the localized $H^{α/2}$-norm of the approximated solution, where $α\in [1,2)$. We demonstrate that the scheme converges strongly in $L^2(0,T;L^2_{loc}(\mathbb{R}))$ to a weak solution of the fractional KdV equation provided the initial data in $L^2(\mathbb{R})$. Assuming the initial data is sufficiently regular, we obtain the rate of convergence for the numerical scheme. Finally, the theoretical convergence rates are justified numerically through various numerical illustrations.

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