arXiv subjects

Gunilla Kreiss

Publications and source records attributed to Gunilla Kreiss.

26 records · Page 2Linked to original sources

arXiv2017

Convergence of finite difference methods for the wave equation in two space dimensions

When using a finite difference method to solve an initial--boundary--value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many previous literatures for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis.

math.NA↗

arXiv2017

Temporal upscaling in micro magnetism via heterogeneous multiscale methods

We consider a multiscale strategy addressing the disparate scales in the Landau-Lifschitz equations in micro-magnetism. At the microscopic scale, the dynamics of magnetic moments are driven by a high frequency field. On the macroscopic scale we are interested in simulating the dynamics of the magnetisation without fully resolving the microscopic scales. The method follows the framework of heterogeneous multiscale methods and it has two main ingredients: a micro- and a macroscale model. The microscopic model is assumed to be known exactly whereas the macro model is incomplete as it lacks effective quantities. The two models use different temporal and spatial scales and effective parameter values for the macro model are computed on the fly, allowing for improved efficiency over traditional one-scale schemes. For the analysis, we consider a single spin under a high frequency field and show that effective quantities can be obtained accurately with step-sizes much larger than the size of the microscopic scales required to resolve the microscopic features. Numerical results both for a single magnetic particle as well as a chain of interacting magnetic particles are given to validate the theory.

math.NA↗

arXiv2017

Atomistic-continuum multiscale modelling of magnetisation dynamics at non-zero temperature

In this article, a few problems related to multiscale modelling of magnetic materials at finite temperatures and possible ways of solving these problems are discussed. The discussion is mainly centred around two established multiscale concepts: the partitioned domain and the upscaling-based methodologies. The major challenge for both multiscale methods is to capture the correct value of magnetisation length accurately, which is affected by a random temperature-dependent force. Moreover, general limitations of these multiscale techniques in application to spin systems are discussed.

physics.comp-ph↗

arXiv2015

Elastic wave propagation in complex geometries: A qualitative comparison between two high order finite difference methods

We compare two high order finite-difference methods that solve the elastic wave equation in two dimensional domains with curved boundaries and material discontinuities. Two numerical experiments are designed with focus on wave boundary interaction, the response of a pressure wave impinging on a circular cavity and the wave field generated by an explosive impact on the wall an underground tunnel. Qualitative comparisons of the results are made where similarities and differences are pointed out.

math.NA↗

arXiv2015

Towards Accurate Modeling of Moving Contact Lines

A main challenge in numerical simulations of moving contact line problems is that the adherence, or no-slip boundary condition leads to a non-integrable stress singularity at the contact line. In this report we perform the first steps in developing the macroscopic part of an accurate multiscale model for a moving contact line problem in two space dimensions. We assume that a micro model has been used to determine a relation between the contact angle and the contact line velocity. An intermediate region is introduced where an analytical expression for the velocity exists. This expression is used to implement boundary conditions for the moving contact line at a macroscopic scale, along a fictitious boundary located a small distance away from the physical boundary. Model problems where the shape of the interface is constant thought the simulation are introduced. For these problems, experiments show that the errors in the resulting contact line velocities converge with the grid size $h$ at a rate of convergence $p\approx 2$.

physics.flu-dyn↗

arXiv2015

High order finite difference methods for the wave equation with non-conforming grid interfaces

We use high order finite difference methods to solve the wave equation in the second order form. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. The focus of this work is on the numerical treatment of non-conforming grid interfaces. The interface conditions are imposed weakly by the simultaneous approximation term technique in combination with interface operators, which move the discrete solutions between the grids on the interface. In particular, we consider interpolation operators and projection operators. A norm-compatibility condition, which leads to stability for first order hyperbolic systems, does not suffice for second order wave equations. An extra constraint on the interface operators must be satisfied to derive an energy estimate for stability. We carry out eigenvalue analyses to investigate the additional constraint and how it is related to stability, and find that the projection operators have better stability properties than the interpolation operators. In addition, a truncation error analysis is performed to study the convergence property of the numerical schemes. In the numerical experiments, the stability and accuracy properties of the numerical schemes are further explored, and the practical usefulness of non-conforming grid interfaces is presented and discussed in two efficiency studies.

math.NA↗

arXiv2015

Convergence of summation-by-parts finite difference methods for the wave equation

In this paper, we consider finite difference approximations of the second order wave equation. We use finite difference operators satisfying the summation-by-parts property to discretize the equation in space. Boundary conditions and grid interface conditions are imposed by the simultaneous-approximation-term technique. Typically, the truncation error is larger at the grid points near a boundary or grid interface than that in the interior. Normal mode analysis can be used to analyze how the large truncation error affects the convergence rate of the underlying stable numerical scheme. If the semi-discretized equation satisfies a determinant condition, two orders are gained from the large truncation error. However, many interesting second order equations do not satisfy the determinant condition. We then carefully analyze the solution of the boundary system to derive a sharp estimate for the error in the solution and acquire the gain in convergence rate. The result shows that stability does not automatically yield a gain of two orders in convergence rate. The accuracy analysis is verified by numerical experiments.

math.NA↗

arXiv2003

Bounds for the threshold amplitude for plane Couette flow

We prove nonlinear stability for finite amplitude perturbations of plane Couette flow. A bound of the solution of the resolvent equation in the unstable complex half-plane is used to estimate the solution of the full nonlinear problem.The result is a lower bound, including Reynolds number dependence,of the threshold amplitude below which all perturbations are stable. Our result is an improvement of the corresponding bound derived in \cite{KLH:1}.

math.AP↗