Related papers: Numerical and approximate solutions for two-dimens…
In this work numerical methods for solving Einstein's equations are developed and applied to the study of inhomogeneous cosmological models. A two-dimensional computer code is described which implements two advanced numerical methods:…
A new multidimensional simulation code for relativistic two-fluid electrodynamics (RTFED) is described. The basic equations consist of the full set of Maxwell's equations coupled with relativistic hydrodynamic equations for separate two…
Multiresolution provides a fundamental tool based on the wavelet theory to build adaptive numerical schemes for Partial Differential Equations and time-adaptive meshes, allowing for error control. We have introduced this strategy before to…
Wavelet theory has been well studied in recent decades. Due to their appealing features such as sparse multiscale representation and fast algorithms, wavelets have enjoyed many tremendous successes in the areas of signal/image processing…
This paper provides a methodology of verified computing for solutions to 1-dimensional advection equations with variable coefficients. The advection equation is typical partial differential equations (PDEs) of hyperbolic type. There are few…
The lack of smoothness is a common feature of weak solutions of nonlinear hyperbolic equations and is a crucial issue in their approximation. This has motivated several efforts to define appropriate indicators, based on the values of the…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
In this work we present a novel bulk-surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk-surface partial differential equations (BSPDEs) in three space dimensions. The BSVEM is based on the discretisation…
The Hamiltonian coupled-mode theory (HCMT), recently derived by Athanassoulis and Papoutsellis [1], provides an efficient new approach for solving fully nonlinear water-wave problems over arbitrary bathymetry. In HCMT, heavy use is made of…
It is well known that closed-form analytical solutions for AC power flow equations do not exist in general. This paper proposes a multi-dimensional holomorphic embedding method (MDHEM) to obtain an explicit approximate analytical AC…
The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary…
The image quality of the new generation of earthbound Extremely Large Telescopes (ELTs) is heavily influenced by atmospheric turbulences. To compensate these optical distortions a technique called adaptive optics (AO) is used. Many AO…
The Holomorphic Embedding Load flow Method (HELM) employs complex analysis to solve the load flow problem. It guarantees finding the correct solution when it exists, and identifying when a solution does not exist. The method, however, is…
In this paper, we develop a computational multiscale to solve the parabolic wave approximation with heterogeneous and variable media. Parabolic wave approximation is a technique to approximate the full wave equation. One benefit of the…
A new fully discrete linearized $H^1$-conforming Lagrange finite element method is proposed for solving the two-dimensional magneto-hydrodynamics equations based on a magnetic potential formulation. The proposed method yields numerical…
Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored…
We propose and analyze an efficient spectral-Galerkin approximation for the Maxwell transmission eigenvalue problem in spherical geometry. Using a vector spherical harmonic expansion, we reduce the problem to a sequence of equivalent…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
We propose a numerical method to solve parameter-dependent hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of…
We consider the 2+1 and 3+1 scalar wave equations reduced via a helical Killing field, respectively referred to as the 2-dimensional and 3-dimensional helically reduced wave equation (HRWE). The HRWE serves as the fundamental model for the…