| 1. |
- af Klinteberg, Ludvig, et al.
(author)
-
Error estimation for quadrature by expansion in layer potential evaluation
- 2017
-
In: Advances in Computational Mathematics. - : Springer. - 1019-7168 .- 1572-9044. ; 43:1, s. 195-234
-
Journal article (peer-reviewed)abstract
- In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.
|
|
| 2. |
- Arjmand, Doghonay, et al.
(author)
-
Atomistic-continuum multiscale modelling of magnetisation dynamics at non-zero temperature
- 2018
-
In: Advances in Computational Mathematics. - : Springer Science and Business Media LLC. - 1019-7168 .- 1572-9044. ; 44:4, s. 1119-1151
-
Journal article (peer-reviewed)abstract
- 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.
|
|
| 3. |
- Arjmand, Doghonay, et al.
(author)
-
Modelling long-range interactions in multiscale simulations of ferromagnetic materials
- 2020
-
In: Advances in Computational Mathematics. - New York : Springer. - 1019-7168 .- 1572-9044. ; 46:1
-
Journal article (peer-reviewed)abstract
- Atomistic-continuum multiscale modelling is becoming an increasingly popular tool for simulating the behaviour of materials due to its computational efficiency and reliable accuracy. In the case of ferromagnetic materials, the atomistic approach handles the dynamics of spin magnetic moments of individual atoms, while the continuum approximations operate with volume-averaged quantities, such as magnetisation. One of the challenges for multiscale models in relation to physics of ferromagnets is the existence of the long-range dipole-dipole interactions between spins. The aim of the present paper is to demonstrate a way of including these interactions into existing atomistic-continuum coupling methods based on the partitioned-domain and the upscaling strategies. This is achieved by modelling the demagnetising field exclusively at the continuum level and coupling it to both scales. Such an approach relies on the atomistic expression for the magnetisation field converging to the continuum expression when the interatomic spacing approaches zero, which is demonstrated in this paper.
|
|
| 4. |
- Asadzadeh, Mohammad, 1952, et al.
(author)
-
Convergence analysis for Backward-Euler and mixed discontinuous Galerkin methods for the Vlasov-Poisson system .
- 2015
-
In: Advances in Computational Mathematics. - : Springer Science and Business Media LLC. - 1019-7168 .- 1572-9044. ; 41:4, s. 833-852
-
Journal article (peer-reviewed)abstract
- We construct and analyze a numerical scheme for the two-dimensional Vlasov-Poisson system based on a backward-Euler (BE) approximation in time combined with a mixed finite element method for a discretization of the Poisson equation in the spatial domain and a discontinuous Galerkin (DG) finite element approximation in the phase-space variables for the Vlasov equation. We prove the stability estimates and derive the optimal convergence rates depending upon the compatibility of the finite element meshes, used for the discretizations of the spatial variable in Poisson (mixed) and Vlasov (DG) equations, respectively. The error estimates for the Poisson equation are based on using Brezzi-Douglas-Marini (BDM) elements in L 2 and H −s , s>0, norms.
|
|
| 5. |
- Blom, David S., et al.
(author)
-
A comparison of Rosenbrock and ESDIRK methods combined with iterative solvers for unsteady compressible flows
- 2016
-
In: Advances in Computational Mathematics. - : Springer Science and Business Media LLC. - 1572-9044 .- 1019-7168. ; 42:6, s. 1401-1426
-
Journal article (peer-reviewed)abstract
- In this article, we endeavour to find a fast solver for finite volume discretizations for compressible unsteady viscous flows. Thereby, we concentrate on comparing the efficiency of important classes of time integration schemes, namely time adaptive Rosenbrock, singly diagonally implicit (SDIRK) and explicit first stage singly diagonally implicit Runge-Kutta (ESDIRK) methods. To make the comparison fair, efficient equation system solvers need to be chosen and a smart choice of tolerances is needed. This is determined from the tolerance TOL that steers time adaptivity. For implicit Runge-Kutta methods, the solver is given by preconditioned inexact Jacobian-free Newton-Krylov (JFNK) and for Rosenbrock, it is preconditioned Jacobian-free GMRES. To specify the tolerances in there, we suggest a simple strategy of using TOL/100 that is a good compromise between stability and computational effort. Numerical experiments for different test cases show that the fourth order Rosenbrock method RODASP and the fourth order ESDIRK method ESDIRK4 are best for fine tolerances, with RODASP being the most robust scheme.
|
|
| 6. |
- Chen, C. C., et al.
(author)
-
Drift-preserving numerical integrators for stochastic Hamiltonian systems
- 2020
-
In: Advances in Computational Mathematics. - : Springer Science and Business Media LLC. - 1019-7168 .- 1572-9044. ; 46:2
-
Journal article (peer-reviewed)abstract
- The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.
|
|
| 7. |
- Farazandeh, Elham, et al.
(author)
-
A rational RBF interpolation with conditionally positive definite kernels
- 2021
-
In: Advances in Computational Mathematics. - : Springer Nature. - 1019-7168 .- 1572-9044. ; 47:5
-
Journal article (peer-reviewed)abstract
- In this paper, we present a rational RBF interpolation method to approximate multivariate functions with poles or other singularities on or near the domain of approximation. The method is based on scattered point layouts and is flexible with respect to the geometry of the problem’s domain. Despite the existing rational RBF-based techniques, the new method allows the use of conditionally positive definite kernels as basis functions. In particular, we use polyharmonic kernels and prove that the rational polyharmonic interpolation is scalable. The scaling property results in a stable algorithm provided that the method be implemented in a localized form. To this aim, we combine the rational polyharmonic interpolation with the partition of unity method. Sufficient number of numerical examples in one, two and three dimensions are given to show the efficiency and the accuracy of the method.
|
|
| 8. |
- Feng, Tao, et al.
(author)
-
Adaptive finite element methods for the identification of distributed parameters in elliptic equation
- 2008
-
In: Advances in Computational Mathematics. - : Springer Science and Business Media LLC. - 1019-7168 .- 1572-9044. ; 29:1, s. 27-53
-
Journal article (peer-reviewed)abstract
- In this paper, adaptive finite element method is developed for the estimation of distributed parameter in elliptic equation. Both upper and lower error bound are derived and used to improve the accuracy by appropriate mesh refinement. An efficient preconditioned project gradient algorithm is employed to solve the nonlinear least-squares problem arising in the context of parameter identification problem. The efficiency of our error estimators is demonstrated by some numerical experiments
|
|
| 9. |
- Fryklund, Fredrik, 1988-, et al.
(author)
-
An adaptive kernel-split quadrature method for parameter-dependent layer potentials
- 2022
-
In: Advances in Computational Mathematics. - : Springer Nature. - 1019-7168 .- 1572-9044. ; 48:2
-
Journal article (peer-reviewed)abstract
- Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer potentials belonging to the modified Helmholtz, modified biharmonic, and modified Stokes equations. These equations depend on a parameter, denoted alpha, and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the source geometry. The refinement is carried out through recursive bisection, with a carefully selected rule set. This maintains accuracy for a wide range of the parameter alpha, at an increased cost that scales as log alpha. Using this algorithm allows kernel-split quadrature to be both accurate and efficient for a much wider range of problems than previously possible.
|
|
| 10. |
- Fryklund, Fredrik, et al.
(author)
-
An integral equation-based numerical method for the forced heat equation on complex domains
- 2020
-
In: Advances in Computational Mathematics. - : Springer Nature. - 1019-7168 .- 1572-9044. ; 46:5
-
Journal article (peer-reviewed)abstract
- Integral equation-based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.
|
|
