Related papers: Wavelet methods to eliminate resonances in the Gal…
It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wavenumbers in excess of a threshold $K_G$ exhibit unexpected features. The study is carried out for both the…
We present numerical simulations of the three-dimensional Galerkin truncated incompressible Euler equations that we integrate in time while regularizing the solution by applying a wavelet-based denoising. For this, at each time step, the…
The Helmholtz equation with variable wavenumbers is challenging to solve numerically due to the pollution effect, which often results in a huge ill-conditioned linear system. In this paper, we present a high-order wavelet Galerkin method to…
Finite-dimensional, inviscid equations of hydrodynamics, such as the zero-viscosity, one-dimensional Burgers equation or the three-dimensional incompressible Euler equation, obtained through a Fourier-Galerkin projection,…
Within the class of nonlinear hyperbolic balance laws posed on a curved spacetime (endowed with a volume form), we identify a hyperbolic balance law that enjoys the same Lorentz invariance property as the one satisfied by the Euler…
We consider the semilinear stochastic heat equation perturbed by additive noise. After time-discretization by Euler's method the equation is split into a linear stochastic equation and a non-linear random evolution equation. The linear…
The solution $u$ of an elliptic interface problem in a domain $\Omega$ is often smooth away from the interface $\Gamma\subset \Omega$, but its gradient is discontinuous across $\Gamma$, resulting in low regularity; in particular, $u \notin…
We propose a high-order spacetime wavelet method for the solution of nonlinear partial differential equations with a user-prescribed accuracy. The technique utilizes wavelet theory with a priori error estimates to discretize the problem in…
This paper presents a wavelet Galerkin method for solving elliptic interface problems of the form $-\nabla\cdot(a\nabla u)=f$ in $\Omega\backslash \Gamma$, where $\Gamma$ is a smooth interface within $\Omega$. Since the scalar variable…
We propose and analyze a space-time finite element method for Westervelt's quasilinear model of ultrasound waves in second-order formulation. The method combines conforming finite element spatial discretizations with a…
We propose a meshless conservative Galerkin method for solving Hamiltonian wave equations. We first discretize the equation in space using radial basis functions in a Galerkin-type formulation. Differ from the traditional RBF Galerkin…
I discuss approaches to optimally remove noise from images. A generalization of Wiener filtering to Non-Gaussian distributions and wavelets is described, as well as an approach to measure the errors in the reconstructed images. We argue…
We construct global-in-time weak solutions to the pressureless Euler alignment system posed on the whole line and supplemented with initial conditions, where an initial density is an arbitrary, nonnegative, bounded, and integrable function…
We present a Fourier--Galerkin framework for the analysis and computation of subwavelength resonances in two-dimensional scattering problems in finite domains. Starting from the boundary integral formulation, we project the operator onto…
We present a new method of wavelet packet decomposition to be used in gravitational wave detection. An issue in wavelet analysis is what is the time-frequency resolution which is best suited to analyze data when in quest of a signal of…
Uncertainty Quantification for nonlinear hyperbolic problems becomes a challenging task in the vicinity of shocks. Standard intrusive methods lead to oscillatory solutions and can result in non-hyperbolic moment systems. The intrusive…
In astrophysics, the two main methods traditionally in use for solving the Euler equations of ideal fluid dynamics are smoothed particle hydrodynamics and finite volume discretization on a stationary mesh. However, the goal to efficiently…
In this article a simplified weak Galerkin finite element method is developed for the Dirichlet boundary value problem of convection-diffusion-reaction equations. The simplified weak Galerkin method utilizes only the degrees of freedom on…
In this paper, we study both convergence and bounded variation properties of a new fully discrete conservative Lagrangian--Eulerian scheme to the entropy solution in the sense of Kruzhkov (scalar case) by using a weak asymptotic analysis.…
The technique of complex scaling for time harmonic wave type equations relies on a complex coordinate stretching to generate exponentially decaying solutions. In this work, we use a Galerkin method with ansatz functions with infinite…