Search results for " Numerical analysis"
showing 6 items of 106 documents
A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries
2021
Author(s): Gulizzi, Vincenzo; Almgren, Ann S; Bell, John B | Abstract: We present a computational framework for solving the equations of inviscid gas dynamics using structured grids with embedded geometries. The novelty of the proposed approach is the use of high-order discontinuous Galerkin (dG) schemes and a shock-capturing Finite Volume (FV) scheme coupled via an $hp$ adaptive mesh refinement ($hp$-AMR) strategy that offers high-order accurate resolution of the embedded geometries. The $hp$-AMR strategy is based on a multi-level block-structured domain partition in which each level is represented by block-structured Cartesian grids and the embedded geometry is represented implicitly by a…
On a nonlinear Schrödinger equation for nucleons in one space dimension
2021
We study a 1D nonlinear Schrödinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied numerically via the time evolution of perturbed ground states. In the time evolution of general localized initial data, they are shown to appear in the long time behaviour of certain cases.
Functional a posteriori error estimates for boundary element methods
2019
Functional error estimates are well-established tools for a posteriori error estimation and related adaptive mesh-refinement for the finite element method (FEM). The present work proposes a first functional error estimate for the boundary element method (BEM). One key feature is that the derived error estimates are independent of the BEM discretization and provide guaranteed lower and upper bounds for the unknown error. In particular, our analysis covers Galerkin BEM and the collocation method, what makes the approach of particular interest for scientific computations and engineering applications. Numerical experiments for the Laplace problem confirm the theoretical results.
Numerical methods for a nonlinear impact model: A comparative study with closed-form corrections
2011
A physically based impact model-already known and exploited in the field of sound synthesis-is studied using both analytical tools and numerical simulations. It is shown that the Hamiltonian of a physical system composed of a mass impacting on a wall can be expressed analytically as a function of the mass velocity during contact. Moreover, an efficient and accurate approximation for the mass outbound velocity is presented, which allows to estimate the Hamiltonian at the end of the contact. Analytical results are then compared to numerical simulations obtained by discretizing the system with several numerical methods. It is shown that, for some regions of the parameter space, the trajectorie…
Numerical Recovery of Source Singularities via the Radiative Transfer Equation with Partial Data
2013
The inverse source problem for the radiative transfer equation is considered, with partial data. Here we demonstrate numerical computation of the normal operator $X_{V}^{*}X_{V}$ where $X_{V}$ is the partial data solution operator to the radiative transfer equation. The numerical scheme is based in part on a forward solver designed by F. Monard and G. Bal. We will see that one can detect quite well the visible singularities of an internal optical source $f$ for generic anisotropic $k$ and $\sigma$, with or without noise added to the accessible data $X_{V}f$. In particular, we use a truncated Neumann series to estimate $X_{V}$ and $X_{V}^{*}$, which provides a good approximation of $X_{V}^{*…
A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne-Weinberger inequality
2015
We consider evolutionary reaction-diffusion problem with mixed Dirichlet--Robin boundary conditions. For this class of problems, we derive two-sided estimates of the distance between any function in the admissible energy space and exact solution of the problem. The estimates (majorants and minorants) are explicitly computable and do not contain unknown functions or constants. Moreover, it is proved that the estimates are equivalent to the energy norm of the deviation from the exact solution.