| 1. |
- Wang, Siyang, et al.
(författare)
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CONVERGENCE OF FINITE DIFFERENCE METHODS FOR THE WAVE EQUATION IN TWO SPACE DIMENSIONS
- 2018
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 87:314, s. 2737-2763
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Tidskriftsartikel (refereegranskat)abstract
- 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 research works 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.
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| 2. |
- Ludvigsson, Gustav, et al.
(författare)
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High-Order Numerical Methods for 2D Parabolic Problems in Single and Composite Domains
- 2018
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 76:2, s. 812-847
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.
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| 3. |
- Appelö, Daniel, et al.
(författare)
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Perfectly matched layers for hyperbolic systems: General formulation, well-posedness and stability
- 2006
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Ingår i: SIAM Journal on Applied Mathematics. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1399 .- 1095-712X. ; 67:1, s. 1-23
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Tidskriftsartikel (refereegranskat)abstract
- Since its introduction the perfectly matched layer (PML) has proven to be an accurate and robust method for domain truncation in computational electromagnetics. However, the mathematical analysis of PMLs has been limited to special cases. In particular, the basic question of whether or not a stable PML exists for arbitrary wave propagation problems remains unanswered. In this work we develop general tools for constructing PMLs for first order hyperbolic systems. We present a model with many parameters, which is applicable to all hyperbolic systems and which we prove is well-posed and perfectly matched. We also introduce an automatic method for analyzing the stability of the model and establishing energy inequalities. We illustrate our techniques with applications to Maxwell's equations, the linearized Euler equations, and arbitrary 2 x 2 systems in (2 + 1) dimensions.
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| 4. |
- Kreiss, Gunilla, et al.
(författare)
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Stability of viscous shocks on finite intervals
- 2008
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 187:1, s. 157-183
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Tidskriftsartikel (refereegranskat)abstract
- Consider the Cauchy problem for a system of viscous conservation laws with a solution consisting of a thin, viscous shock layer connecting smooth regions. We expect the time-dependent behavior of such a solution to involve two processes. One process consists of the large-scale evolution of the solution. This process is well modeled by the corresponding inviscid equations. The other process is the adjustment in shape and position of the shock layer to the large-scale solution. The time scale of the second process is much faster than the first, 1/nu compared to 1. The second process can be divided into two parts, adjustment of the shape and of the position. During this adjustment the end states are essentially constant. In order to answer the question of stability we have developed a technique where the two processes can be separated. To isolate the fast process, we consider the region in the vicinity of the shock layer. The equations are augmented with special boundary conditions that reflect the slow change of the end states. We show that, for the isolated fast process, the perturbations decay exponentially in time.
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| 5. |
- Kreiss, Gunilla, et al.
(författare)
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Elimination of first order errors in shock calculations
- 2001
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Ingår i: SIAM Journal on Numerical Analysis. - : SIAM PUBLICATIONS. - 0036-1429 .- 1095-7170. ; 38:6, s. 1986-1998
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Tidskriftsartikel (refereegranskat)abstract
- First order errors downstream of shocks have been detected in computations with higher order shock capturing schemes in one and two dimensions. Based on a matched asymptotic expansion analysis we show how to modify the artificial viscosity and raise the order of accuracy.
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| 6. |
- Weber, Ivy, 1995-, et al.
(författare)
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Stability analysis of high order methods for the wave equation
- 2022
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 404
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we investigate the stability of a numerical method for solving the wave equation. The method uses explicit leap-frog in time and high order continuous and discontinuous (DG) finite elements using the standard Lagrange and Hermite basis functions in space. Matrix eigenvalue analysis is used to calculate time-step restrictions. We show that the time-step restriction for continuous Lagrange elements is independent of the nodal distribution, such as equidistributed Lagrange nodes and Gauss–Lobatto nodes. We show that the time-step restriction for the symmetric interior penalty DG schemes with the usual penalty terms is tighter than for continuous Lagrange finite elements. Finally, we conclude that the best time-step restriction is obtained for continuous Hermite finite elements up to polynomial degrees.
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| 7. |
- Fogelklou, Oswald, 1975-
(författare)
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Computer-Assisted Proofs and Other Methods for Problems Regarding Nonlinear Differential Equations
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This PhD thesis treats some problems concerning nonlinear differential equations. In the first two papers computer-assisted proofs are used. The differential equations there are rewritten as fixed point problems, and the existence of solutions are proved. The problem in the first paper is one-dimensional; with one boundary condition given by an integral. The problem in the second paper is three-dimensional, and Dirichlet boundary conditions are used. Both problems have their origins in fluid dynamics. Paper III describes an inverse problem for the heat equation. Given the solution, a solution dependent diffusion coefficient is estimated by intervals at a finite set of points. The method includes the construction of set-valued level curves and two-dimensional splines. In paper IV we prove that there exists a unique, globally attracting fixed point for a differential equation system. The differential equation system arises as the number of peers in a peer-to-peer network, which is described by a suitably scaled Markov chain, goes to infinity. In the proof linearization and Dulac's criterion are used.
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| 8. |
- Zahedi, Sara, et al.
(författare)
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A Conservative Level Set Method for Contact Line Dynamics
- 2009
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 228:17, s. 6361-6375
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Tidskriftsartikel (refereegranskat)abstract
- A new model for simulating contact line dynamics is proposed. We apply the idea of driving contact-line movement by enforcing the equilibrium contact angle at the boundary, to the conservative level set method for incompressible two-phase flow [E. Olsson, G. Kreiss, A conservative level set method for two phase flow, J. Comput. Phys. 210 (2005) 225-246]. A modified reinitialization procedure provides a diffusive mechanism for contact-line movement, and results in a smooth transition of the interface near the contact line without explicit reconstruction of the interface. We are able to capture contact-line movement without loosing the conservation. Numerical simulations of capillary dominated flows in two space dimensions demonstrate that the model is able to capture contact line dynamics qualitatively correct.
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| 9. |
- Appelö, Daniel, et al.
(författare)
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An explicit Hermite–Taylor method for the Schrödinger equation
- 2017
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Ingår i: Communications in Computational Physics. - : Global Science Press. - 1815-2406 .- 1991-7120. ; 21, s. 1207-1230
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Tidskriftsartikel (refereegranskat)abstract
- An explicit spectrally accurate order-adaptive Hermite-Taylor method for the Schrödinger equation is developed. Numerical experiments illustrating the properties of the method are presented. The method, which is able to use very coarse grids while still retaining high accuracy, compares favorably to an existing exponential integrator – high order summation-by-parts finite difference method.
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| 10. |
- Arjmand, Doghonay, et al.
(författare)
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An equation-free approach for second order multiscale hyperbolic problems in non-divergence form
- 2018
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Ingår i: Communications in Mathematical Sciences. - 1539-6746 .- 1945-0796. ; 16:8, s. 2317-2343
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Tidskriftsartikel (refereegranskat)abstract
- The present study concerns the numerical homogenization of second order hyperbolic equations in non-divergence form, where the model problem includes a rapidly oscillating coefficient function. These small scales influence the large scale behavior, hence their effects should be accurately modelled in a numerical simulation. A direct numerical simulation is prohibitively expensive since a minimum of two points per wavelength are needed to resolve the small scales. A multiscale method, under the equation-free methodology, is proposed to approximate the coarse scale behaviour of the exact solution at a cost independent of the small scales in the problem. We prove convergence rates for the upscaled quantities in one as well as in multi-dimensional periodic settings. Moreover, numerical results in one and two dimensions are provided to support the theory.
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