Mukul Dwivedi

Mukul Dwivedi, Andreas Rupp

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.

Mukul Dwivedi, Jesse Railo, Andreas Rupp

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.

Mukul Dwivedi, Andreas Rupp

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.