Related papers: On the rotating nonlinear Klein-Gordon equation wi…
We introduce a hybrid method to couple continuous Galerkin finite element methods and high-order finite difference methods in a nonconforming multiblock fashion. The aim is to optimize computational efficiency when complex geometries are…
We study the convergence of a finite volume method based on the method of bicharacteristics for multidimensional hyperbolic conservation laws. In particular, we concentrate on the linear wave equation system and nonlinear Euler equations of…
The robustness and accuracy of marginally resolved discontinuous Galerkin spectral element computations are evaluated for the standard formulation and a kinetic energy conserving split form on complex flow problems of physical and…
In the framework of the relativistic generalized uncertainty principle (RGUP), we study the nonlinear Klein-Gordon field with $\phi^4$ self-interaction. This generalization comes from the quantum gravity theories that predict the existence…
The proximal Galerkin finite element method is a high-order, low-iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of point-wise bound constraints in infinite-dimensional function spaces.…
In this article we consider a system of two Klein-Gordon equations, set on the $d$-dimensional box of size $L$, coupled through quadratic semilinear terms of strength $\varepsilon$ and evolving from well-prepared random initial data. We…
We propose and analyze structure-preserving parametric finite element methods (SP-PFEM) for evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy $\gamma(\boldsymbol{n})$ for…
We study soliton solutions to the Klein-Gordon equation via Lie symmetries and the travelling-wave ansatz. It is shown, by taking a linear combination of the spatial and temporal Lie point symmetries, that soliton solutions naturally exist,…
We consider the one-constant Landau - de Gennes model for nematic liquid crystals. The order parameter is a traceless tensor field $\mathbf{Q}$, which is constrained to be uniaxial: $\mathbf{Q} = s (\mathbf{n}\otimes\mathbf{n} - d^{-1}…
When nonconforming discontinuous Galerkin methods are implemented for hyperbolic equations using quadrature, exponential energy growth can result even when the underlying scheme with exact integration does not support such growth. Using…
In this paper, we propose a neural multiscale decomposition method (NeuralMD) for solving the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter $\varepsilon\in(0,1]$ from the relativistic regime to the nonrelativistic…
We study the 1D Klein-Gordon equation with variable coefficient cubic nonlinearity. This problem exhibits a striking resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the…
We develop the general form of the variational multiscale method in a discontinuous Galerkin framework. Our method is based on the decomposition of the true solution into discontinuous coarse-scale and discontinuous fine-scale parts. The…
The dynamics of vortex solitons in a BEC superfluid is studied. A quantum lattice-gas algorithm (localization-based quantum computation) is employed to examine the dynamical behavior of vortex soliton solutions of the Gross-Pitaevskii…
We study the dynamics of a bounded gravitational collapsing configuration emitting gravitational waves, where the exterior spacetime is described by Robinson-Trautman geometries. The full nonlinear regime is examined by using the Galerkin…
In this paper, we develop and analyze a novel numerical scheme for the steady incompressible Navier-Stokes equations by the weak Galerkin methods. The divergence-preserving velocity reconstruction operator is employed in the discretization…
We develop and analyze a class of structure-preserving discontinuous Galerkin schemes for the nonlinear Vlasov-Poisson-Fokker-Planck model, reformulated as a hyperbolic system through a Hermite expansion in the velocity variable. We…
In this paper, we develop bound-preserving techniques for the Runge--Kutta (RK) discontinuous Galerkin (DG) method with compact stencils (cRKDG method) for hyperbolic conservation laws. The cRKDG method was recently introduced in [Q. Chen,…
In this paper, we investigate the almost-periodic solutions for the one-dimensional nonlinear Klein-Gordon equation within the non-relativistic limit under periodic boundary conditions. Specifically, by employing the method introduced in…
We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical…