arXiv subjects

Daniel Appelö

Publications and source records attributed to Daniel Appelö.

30 records · Page 2Linked to original sources

arXiv2022

An energy-based summation-by-parts finite difference method for the wave equation in second order form

We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the simultaneous-approximation-term (SAT) method, we derive energy estimates for the semi-discretization. In addition, error estimates are derived by the normal mode analysis. The proposed method is termed as energy-based because of its similarity with the energy-based discontinuous Galerkin method. When imposing the Dirichlet boundary condition and material interface conditions, the traditional SBP-SAT discretization uses a penalty term with a mesh-dependent parameter, which is not needed in our method. Furthermore, numerical dissipation can be added to the discretization through the boundary and interface conditions. We present numerical experiments that verify convergence and robustness of the proposed method.

math.NA↗

arXiv2022

EM-WaveHoltz: A flexible frequency-domain method built from time-domain solvers

A novel approach to computing time-harmonic solutions of Maxwell's equations by time-domain simulations is presented. The method, EM-WaveHoltz, results in a positive definite system of equations which makes it amenable to iterative solution with the conjugate gradient method or with GMRES. Theoretical results guaranteeing the convergence of the method away from resonances is presented. Numerical examples illustrating the properties of EM-WaveHoltz are given.

math.NA↗

arXiv2021

A Hermite Method with a Discontinuity Sensor for Hamilton-Jacobi Equations

We present a Hermite interpolation based partial differential equation solver for Hamilton-Jacobi equations. Many Hamilton-Jacobi equations have a nonlinear dependency on the gradient, which gives rise to discontinuities in the derivatives of the solution, resulting in kinks. We built our solver with two goals in mind: 1) high order accuracy in smooth regions and 2) sharp resolution of kinks. To achieve this, we use Hermite interpolation with a smoothness sensor. The degrees-of freedom of Hermite methods are tensor-product Taylor polynomials of degree $m$ in each coordinate direction. The method uses $(m+1)^d$ degrees of freedom per node in $d$-dimensions and achieves an order of accuracy $(2m+1)$ when the solution is smooth. To obtain sharp resolution of kinks, we sense the smoothness of the solution on each cell at each timestep. If the solution is smooth, we march the interpolant forward in time with no modifications. When our method encounters a cell over which the solution is not smooth, it introduces artificial viscosity locally while proceeding normally in smooth regions. We show through numerical experiments that the solver sharply captures kinks once the solution losses continuity in the derivative while achieving $2m+1$ order accuracy in smooth regions.

math.NA↗

arXiv2021

Anderson Acceleration Based on the $\mathcal{H}^{-s}$ Sobolev Norm for Contractive and Noncontractive Fixed-Point Operators

Anderson acceleration (AA) is a technique for accelerating the convergence of fixed-point iterations. In this paper, we apply AA to a sequence of functions and modify the norm in its internal optimization problem to the $\mathcal{H}^{-s}$ norm, for some positive integer $s$, to bias it towards low-frequency spectral content in the residual. We analyze the convergence of AA by quantifying its improvement over Picard iteration. We find that AA based on the $\mathcal{H}^{-2}$ norm is well-suited to solve fixed-point operators derived from second-order elliptic differential operators, including the Helmholtz equation.

math.NA↗

arXiv2021

Accuracy of spectral element method for wave, parabolic and Schrödinger equations

The spectral element method constructed by the $Q^k$ ($k\geq 2$) continuous finite element method with $(k+1)$-point Gauss-Lobatto quadrature on rectangular meshes is a popular high order scheme for solving wave equations in various applications. It can also be regarded as a finite difference scheme on all Gauss-Lobatto points. We prove that this finite difference scheme is $(k+2)$-order accurate in discrete 2-norm for smooth solutions. The same proof can be extended to the spectral element method solving linear parabolic and Schrödinger equations. The main result also applies to the spectral element method on curvilinear meshes that can be smoothly mapped to rectangular meshes on the unit square.

math.NA↗

arXiv2021

Energy-based discontinuous Galerkin difference methods for second-order wave equations

We combine the newly-constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant.

math.NA↗

arXiv2021

Re-evaluation of Spin-Orbit Dynamics of Polarized e+ e- Beams in High Energy Circular Accelerators and Storage Rings: an approach based on a Bloch equation

We give an overview of our current/future analytical and numerical work on the spin polarization in high-energy electron storage rings. Our goal is to study the possibility of polarization for the CEPC and FCC-ee. Our work is based on the so-called Bloch equation for the polarization density introduced by Derbenev and Kondratenko in 1975. We also give an outline of the standard approach, the latter being based on the Derbenev-Kondratenko formulas.

physics.acc-ph↗

arXiv2020

The Bloch equation for spin dynamics in electron storage rings: computational and theoretical aspects

In this paper we describe our work on spin polarization in high-energy electron storage rings which we base on the Bloch equation for the polarization density and which aims towards the e-/e+ option of the proposed Future Circular Collider (FCC-ee) and the proposed Circular Electron Positron Collider (CEPC). The Bloch equation takes into account non spin-flip and spin-flip effects due to synchrotron radiation including the spin-diffusion effects and the Sokolov-Ternov effect with its Baier-Katkov generalization as well as the kinetic-polarization effect. This mathematical model is an alternative to the standard mathematical model based on the Derbenev-Kondratenko formulas. For our numerical and analytical studies of the Bloch equation we develop an approximation to the latter to obtain an effective Bloch equation. This is accomplished by finding a third mathematical model based on a system of stochastic differential equations underlying the Bloch equation and by approximating that system via the method of averaging from perturbative ODE theory. We also give an overview of our algorithm for numerically integrating the effective Bloch equation. This discretizes the phase space using spectral methods and discretizes time via the additive Runge-Kutta method which is a high-order semi-implicit method. We also discuss the relevance of the third mathematical model for spin tracking.

physics.acc-ph↗

arXiv2020

Hermite-Discontinuous Galerkin Overset Grid Methods for the Scalar Wave Equation

We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with eometrically flexible discontinuous Galerkin methods by using overset grids. Near boundaries we use thin boundary fitted curvilinear grids and in the volume we use Cartesian grids so that the computational complexity of the solvers approach a structured Cartesian Hermite method. Unlike many other overset methods we do not need to add artificial dissipation but we find that the built in dissipation of the Hermite and discontinuous Galerkin methods is sufficient to maintain stability. By numerical experiments we demonstrate the stability, accuracy, efficiency and applicability of the methods to forward and inverse problems.

math.NA↗

arXiv2019

An Energy Based Discontinuous Galerkin Method for Coupled Elasto-Acoustic Wave Equations in Second Order Form

We consider wave propagation in a coupled fluid-solid region, separated by a static but possibly curved interface. The wave propagation is modeled by the acoustic wave equation in terms of a velocity potential in the fluid, and the elastic wave equation for the displacement in the solid. At the fluid solid interface, we impose suitable interface conditions to couple the two equations. We use a recently developed, energy based discontinuous Galerkin method to discretize the governing equations in space. Both energy conserving and upwind numerical fluxes are derived to impose the interface conditions. The highlights of the developed scheme include provable energy stability and high order accuracy. We present numerical experiments to illustrate the accuracy property and robustness of the developed scheme.

math.NA↗

arXiv2017

On the scaling of entropy viscosity in high order methods

In this work, we outline the entropy viscosity method and discuss how the choice of scaling influences the size of viscosity for a simple shock problem. We present examples to illustrate the performance of the entropy viscosity method under two distinct scalings.

math.NA↗

arXiv2017

Flux-conservative Hermite methods for simulation of nonlinear conservation laws

A new class of Hermite methods for solving nonlinear conservation laws is presented. While preserving the high order spatial accuracy for smooth solutions in the existing Hermite methods, the new methods come with better stability properties. Artificial viscosity in the form of the entropy viscosity method is added to capture shocks.

math.NA↗