Related papers: Multi-block simulations in general relativity: hig…
Our aim is to construct high order approximation schemes for general semigroups of linear operators $P_{t},t\geq 0$. In order to do it, we fix a time horizon $T $ and the discretization steps $h_{l}=\frac{T}{n^{l}},l\in \mathbb{N}$ and we…
Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms…
Sparse grids are tailored to the approximation of smooth high-dimensional functions. On a $d$-dimensional tensor product space, the number of grid points is $N = \mathcal O(h^{-1} |\log h|^{d-1})$, where $h$ is a mesh parameter. The…
The efficient solution of sparse, linear systems resulting from the discretization of partial differential equations is crucial to the performance of many physics-based simulations. The algorithmic optimality of multilevel approaches for…
Partial differential equations sometimes have critical points where the solution or some of its derivatives are discontinuous. The simplest example is a discontinuity in the initial condition. It is well known that those decrease the…
Second order accurate Cartesian grid methods have been well developed for interface problems in the literature. However, it is challenging to develop third or higher order accurate methods for problems with curved interfaces and internal…
In this paper a class of higher order finite element methods for the discretization of surface Stokes equations is studied. These methods are based on an unfitted finite element approach in which standard Taylor-Hood spaces on an underlying…
This paper presents an extension of a recently developed high order finite difference method for the wave equation on a grid with non-conforming interfaces. The stability proof of the existing methods relies on the interpolation operators…
Common techniques for the spatial discretisation of PDEs on a macroscale grid include finite difference, finite elements and finite volume methods. Such methods typically impose assumed microscale structures on the subgrid fields, so…
We study non-conforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a…
Various classes of stable finite difference schemes can be constructed to obtain a numerical solution. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In this study,…
A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface…
This paper is concerned with the construction of high order schemes on irregular grids for balance laws, including a discussion of an a-posteriori error indicator based on the numerical entropy production. We also impose well-balancing on…
Finite-difference simulations of fluid dynamics and magnetohydrodynamics generally require an explicit diffusion operator, either to maintain stability by attenuating grid-scale structure, or to implement physical diffusivities such as…
Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and…
This paper extends previous work on finitedifference schemes over staggered grids for infinite-dimensional port-Hamiltonian systems. In the one-dimensional setting, it generalizes the discretization approach originally developed for the…
We consider the finite difference discretization of isotropic elastic wave equations on nonuniform grids. The intended applications are seismic studies, where heterogeneity of the earth media can lead to severe oversampling for simulations…
We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence…
This paper summarizes a research program that has been underway for a decade. The objective is to find a fast and accurate scheme for solving quantum problems which does not involve a Monte Carlo algorithm. We use an alternative strategy…
With the increasing number of components and further miniaturization the mean time between faults in supercomputers will decrease. System level fault tolerance techniques are expensive and cost energy, since they are often based on…