| 1. |
- Wang, Siyang, et al.
(author)
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CONVERGENCE OF FINITE DIFFERENCE METHODS FOR THE WAVE EQUATION IN TWO SPACE DIMENSIONS
- 2018
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In: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 87:314, s. 2737-2763
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Journal article (peer-reviewed)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.
(author)
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High-Order Numerical Methods for 2D Parabolic Problems in Single and Composite Domains
- 2018
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In: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 76:2, s. 812-847
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Journal article (peer-reviewed)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.
(author)
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An explicit Hermite–Taylor method for the Schrödinger equation
- 2017
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In: Communications in Computational Physics. - : Global Science Press. - 1815-2406 .- 1991-7120. ; 21, s. 1207-1230
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Journal article (peer-reviewed)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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| 4. |
- Appelö, Daniel, et al.
(author)
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Perfectly matched layers for hyperbolic systems: General formulation, well-posedness and stability
- 2006
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In: SIAM Journal on Applied Mathematics. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1399 .- 1095-712X. ; 67:1, s. 1-23
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Journal article (peer-reviewed)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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| 5. |
- Kieri, Emil, et al.
(author)
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Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrodinger Equations
- 2015
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In: Advances in Applied Mathematics and Mechanics. - : Global Science Press. - 2070-0733 .- 2075-1354. ; 7:6, s. 687-714
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Journal article (peer-reviewed)abstract
- In the semiclassical regime, solutions to the time-dependent Schrodinger equation for molecular dynamics are highly oscillatory. The number of grid points required for resolving the oscillations may become very large even for simple model problems, making solution on a grid intractable. Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the atomic nuclei are heavy and the potential is smooth. However, when the potential has variations on a small length-scale, quantum phenomena become important. Then asymptotic methods are less accurate. The two classes of methods perform well in different parameter regimes. This opens for hybrid methods, using Gaussian beams where we can and finite differences where we have to. We propose a new method for treating the coupling between the finite difference method and Gaussian beams. The new method reduces the needed amount of overlap regions considerably compared to previous methods, which improves the efficiency.
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| 6. |
- Kreiss, Gunilla, et al.
(author)
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Analysis of stretched grids as buffer zones in simulations of wave propagation
- 2016
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In: Applied Numerical Mathematics. - : Elsevier BV. - 0168-9274 .- 1873-5460. ; 107, s. 1-17
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Journal article (peer-reviewed)abstract
- A zone of increasingly stretched grid is a robust and easy-to-use way to avoid unwanted reflections at artificial boundaries in wave propagating simulations. In such a buffer zone there are two main damping mechanisms, dissipation and under-resolution that turns a traveling wave into an evanescent wave. We present analysis in one and two space dimensions showing that evanescent decay through under-resolution is a very efficient way to damp waves. The analysis is supported by numerical computations.
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| 7. |
- Kreiss, Gunilla, et al.
(author)
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Elimination of first order errors in shock calculations
- 2001
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In: SIAM Journal on Numerical Analysis. - : SIAM PUBLICATIONS. - 0036-1429 .- 1095-7170. ; 38:6, s. 1986-1998
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Journal article (peer-reviewed)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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| 8. |
- Wang, Siyang, et al.
(author)
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Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation
- 2017
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In: Journal of Scientific Computing. - : Springer New York LLC. - 0885-7474 .- 1573-7691. ; 71:1, s. 219-245
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Journal article (peer-reviewed)abstract
- When using a finite difference method to solve a time dependent partial differential equation, the truncation error is often larger at a few grid points near a boundary or grid interface than in the interior. In computations, the observed convergence rate is often higher than the order of the large truncation error. In this paper, we develop techniques for analyzing this phenomenon, and particularly consider the second order wave equation. The equation is discretized by a finite difference operator satisfying a summation by parts property, and the boundary and grid interface conditions are imposed weakly by the simultaneous approximation term method. It is well-known that if the semi-discretized wave equation satisfies the determinant condition, that is the boundary system in Laplace space is nonsingular for all Re(s) ≥ 0 , two orders are gained from the large truncation error localized at a few grid points. By performing a normal mode analysis, we show that many common discretizations do not satisfy the determinant condition at s= 0. We then carefully analyze the error equation to determine the gain in the convergence rate. The result shows that stability does not automatically imply a gain of two orders in the convergence rate. The precise gain can be lower than, equal to or higher than two orders, depending on the boundary condition and numerical boundary treatment. The accuracy analysis is verified by numerical experiments, and very good agreement is obtained.
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| 9. |
- Wang, Siyang, et al.
(author)
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High Order Finite Difference Methods for the Wave Equation with Non-conforming Grid Interfaces
- 2016
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In: Journal of Scientific Computing. - : Springer New York LLC. - 0885-7474 .- 1573-7691. ; 68:3, s. 1002-1028
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Journal article (peer-reviewed)abstract
- 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 numerical treatments of non-conforming grid interfaces and non-conforming mesh blocks. Interface conditions are imposed weakly by the simultaneous approximation term technique in combination with interface operators, which move discrete solutions between grids at an interface. In particular, we consider an interpolation approach and a projection approach with corresponding operators. A norm-compatible condition of the interface operators leads to energy stability for first order hyperbolic systems. By imposing an additional constraint on the interface operators, we derive an energy estimate of the numerical scheme for the second order wave equation. We carry out eigenvalue analyses to investigate the additional constraint and its relation to stability. In addition, a truncation error analysis is performed, and discussed in relation to convergence properties of the numerical schemes. In the numerical experiments, stability and accuracy properties of the numerical scheme are further explored, and the practical usefulness of non-conforming grid interfaces and mesh blocks is discussed in two practical examples. © 2016, Springer Science+Business Media New York.
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| 10. |
- Nikkar, Samira
(author)
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Stable High Order Finite Difference Methods for Wave Propagation and Flow Problems on Deforming Domains
- 2016
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Doctoral thesis (other academic/artistic)abstract
- We construct stable, accurate and efficient numerical schemes for wave propagation and flow problems posed on spatial geometries that are moving, deforming, erroneously described or non-simply connected. The schemes are on Summation-by-Parts (SBP) form, combined with the Simultaneous Approximation Term (SAT) technique for imposing initial and boundary conditions. The main analytical tool is the energy method, by which well-posedness, stability and conservation are investigated. To handle the deforming domains, time-dependent coordinate transformations are used to map the problem to fixed geometries.The discretization is performed in such a way that the Numerical Geometric Conservation Law (NGCL) is satisfied. Additionally, even though the schemes are constructed on fixed domains, time-dependent penalty formulations are necessary, due to the originally moving boundaries. We show how to satisfy the NGCL and present an automatic formulation for the penalty operators, such that the correct number of boundary conditions are imposed, when and where required.For problems posed on erroneously described geometries, we investigate how the accuracy of the solution is affected. It is shown that the inaccurate geometry descriptions may lead to wrong wave speeds, a misplacement of the boundary condition, the wrong boundary operator or a mismatch of data. Next, the SBP-SAT technique is extended to time-dependent coupling procedures for deforming interfaces in hyperbolic problems. We prove conservation and stability and show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the variations of the interface location while the NGCL is preserved.Moreover, dual consistent SBP-SAT schemes for the linearized incompressible Navier-Stokes equations posed on deforming domains are investigated. To simplify the derivations of the dual problem and incorporate the motions of the boundaries, the second order formulation is reduced to first order and the problem is transformed to a fixed domain. We prove energy stability and dual consistency. It is shown that the solution as well as the divergence of the solution converge with the design order of accuracy, and that functionals of the solution are superconverging.Finally, initial boundary value problems posed on non-simply connected spatial domains are investigated. The new formulation increases the accuracy of the scheme by minimizing the use of multi-block couplings. In order to show stability, the spectrum of the semi-discrete SBP-SAT formulation is studied. We show that the eigenvalues have the correct sign, which implies stability, in combination with the SBP-SAT technique in time.
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