| 1. |
- af Klinteberg, Ludvig
(författare)
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Singularity swap quadrature for nearly singular line integrals on closed curves in two dimensions
- 2024
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Ingår i: BIT Numerical Mathematics. - 0006-3835 .- 1572-9125. ; 64:1
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Tidskriftsartikel (refereegranskat)abstract
- This paper presents a quadrature method for evaluating layer potentials in two dimensions close to periodic boundaries, discretized using the trapezoidal rule. It is an extension of the method of singularity swap quadrature, which recently was introduced for boundaries discretized using composite Gauss–Legendre quadrature. The original method builds on swapping the target singularity for its preimage in the complexified space of the curve parametrization, where the source panel is flat. This allows the integral to be efficiently evaluated using an interpolatory quadrature with a monomial basis. In this extension, we use the target preimage to swap the singularity to a point close to the unit circle. This allows us to evaluate the integral using an interpolatory quadrature with complex exponential basis functions. This is well-conditioned, and can be efficiently evaluated using the fast Fourier transform. The resulting method has exponential convergence, and can be used to accurately evaluate layer potentials close to the source geometry. We report experimental results on a simple test geometry, and provide a baseline Julia implementation that can be used for further experimentation.
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| 2. |
- Akrivis, Georgios, et al.
(författare)
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Linearly implicit finite element methods for the time-dependent Joule heating problem
- 2005
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Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 45:3, s. 429-442
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Tidskriftsartikel (refereegranskat)abstract
- Completely discrete numerical methods for a nonlinear elliptic-parabolic system, the time-dependent Joule heating problem, are introduced and analyzed. The equations are discretized in space by a standard finite element method, and in time by combinations of rational implicit and explicit multistep schemes. The schemes are linearly implicit in the sense that they require, at each time level, the solution of linear systems of equations. Optimal order error estimates are proved under the assumption of sufficiently regular solutions.
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| 3. |
- Arévalo, Carmen, et al.
(författare)
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Convergence of multistep discretizations of DAEs
- 1995
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Ingår i: BIT. - 0006-3835. ; 35:2, s. 143-168
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Tidskriftsartikel (refereegranskat)abstract
- Standard ODE methods such as linear multistep methods encounter difficulties when applied to differential-algebraic equations (DAEs) of index greater than 1. In particular, previous results for index 2 DAEs have practically ruled out the use of all explicit methods and of implicit multistep methods other than backward difference formulas (BDFs) because of stability considerations. In this paper we embed known results for semi-explicit index 1 and 2 DAEs in a more comprehensive theory based on compound multistep and one-leg discretizations. This explains and characterizes the necessary requirements that a method must fulfill in order to be applicable to semi-explicit DAEs. Thus we conclude that the most useful discretizations are those that avoid discretization of the constraint. A freer use of e.g. explicit methods for the non-stiff differential part of the DAE is then possible.
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| 4. |
- Arévalo, Carmen, et al.
(författare)
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Improving the accuracy of BDF methods for index 3 dierential algebraic equations
- 1995
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Ingår i: BIT. - 0006-3835 .- 1572-9125. ; 35:3, s. 297-308
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Tidskriftsartikel (refereegranskat)abstract
- Methods for solving index 3 DAEs based on BDFs suffer a loss of accuracy when there is a change of step size or a change of order of the method. A layer of nonuniform convergence is observed in these cases, and O(1) errors may appear in the algebraic variables. From the viewpoint of error control, it is beneficial to allow smooth changes of step size, and since most codes based on BDFs are of variable order, it is also of interest to avoid the inaccuracies caused by a change of order of the method. In the case of BDFs applied to index 3 DAEs in semi-explicit form, we present algorithms that correct to O(h) the inaccurate approximations to the algebraic variables when there are changes of step size in the backward Euler method. These algorithms can be included in an existing code at a very small cost. We have also described how to obtain formulas that correct the O(1) errors in the algebraic variables appearing after a change of order.
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| 5. |
- Arévalo, Carmen, et al.
(författare)
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Stabilized multistep methods for index 2 Euler-Lagrange DAEs
- 1996
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Ingår i: BIT. - 0006-3835. ; 36:1, s. 1-13
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Tidskriftsartikel (refereegranskat)abstract
- We consider multistep discretizations, stabilized by β-blocking, for Euler-Lagrange DAEs of index 2. Thus we may use “nonstiff” multistep methods with an appropriate stabilizing difference correction applied to the Lagrangian multiplier term. We show that order p =k + 1 can be achieved for the differential variables with order p =k for the Lagrangian multiplier fork-step difference corrected BDF methods as well as for low order k-step Adams-Moulton methods. This approach is related to the recently proposed “half-explicit” Runge-Kutta methods.
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| 6. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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A posteriori error estimates for streamline-diffusion and discontinuous Galerkin methods for the Vlasov-Maxwell system
- 2018
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Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 58:1, s. 5-26
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Tidskriftsartikel (refereegranskat)abstract
- This paper concerns a posteriori error analysis for the streamline diffusion (SD) finite element method for the one and one-half dimensional relativistic Vlasov– Maxwell system. The SD scheme yields a weak formulation, that corresponds to an add of extra diffusion to, e.g. the system of equations having hyperbolic nature, or convection-dominated convection diffusion problems. The a posteriori error estimates rely on dual formulations and yield error controls based on the computable residuals. The convergence estimates are derived in negative norms, where the error is split into an iteration and an approximation error and the iteration procedure is assumed to converge.
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| 7. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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A posteriori error estimates for the one and one-half Dimensional Relativistic Vlasov–Maxwell system
- 2018
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Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 58:1, s. 5-26
-
Tidskriftsartikel (refereegranskat)abstract
- © 2017 The Author(s) This paper concerns a posteriori error analysis for the streamline diffusion (SD) finite element method for the one and one-half dimensional relativistic Vlasov–Maxwell system. The SD scheme yields a weak formulation, that corresponds to an add of extra diffusion to, e.g. the system of equations having hyperbolic nature, or convection-dominated convection diffusion problems. The a posteriori error estimates rely on dual formulations and yield error controls based on the computable residuals. The convergence estimates are derived in negative norms, where the error is split into an iteration and an approximation error and the iteration procedure is assumed to converge.
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| 8. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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On a multiwavelet spectral element method for integral equation of a generalized Cauchy problem
- 2022
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Ingår i: Bit Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 62, s. 1383-1416
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we deal with construction and analysis of a multiwavelet spectral element scheme for a generalized Cauchy type problem with Caputo fractional derivative. Numerical schemes for this type of problems, often suffer from the draw-back of spurious oscillations. A common remedy is to render the problem to an equivalent integral equation. For the generalized Cauchy type problem, a corresponding integral equation is of nonlinear Volterra type. In this paper we investigate wellposedness and convergence of a stabilizing multiwavelet scheme for a, one-dimensional case (in [a, b] or [0, 1]), of this problem. Based on multiwavelets, we construct an approximation procedure for the fractional integral operator that yields a linear system of equations with sparse coefficient matrix. In this setting, choosing an appropriate threshold, the number of non-zero coefficients in the system is substantially reduced. A severe obstacle in the convergence analysis is the lack of continuous derivatives in the vicinity of the inflow/ starting boundary point. We overcome this issue through separating a J (mesh)-dependent, small, neighborhood of alpha (or origin) from the interval, where we only take L-2-norm. The estimate in this part relies on Chebyshev polynomials, viz. As reported by Richardson(Chebyshev interpolation for functions with endpoint singularities via exponential and double-exponential transforms, Oxford University, UK, 2012) and decreases, almost, exponentially by raising J. At the remaining part of the domain the solution is sufficiently regular to derive the desired optimal error bound. We construct such a modified scheme and analyze its wellposedness, efficiency and accuracy. The robustness of the proposed scheme is confirmed implementing numerical examples.
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| 9. |
- Barrera, Tony, et al.
(författare)
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An integer based square root algorithm
- 1993
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Ingår i: BIT Numerical Mathematics. - 0006-3835 .- 1572-9125. ; 33, s. 254-261
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Tidskriftsartikel (refereegranskat)
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| 10. |
- Barth, A., et al.
(författare)
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Multilevel Monte Carlo method for parabolic stochastic partial differential equations
- 2013
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Ingår i: BIT (Copenhagen). - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 53:1, s. 3-27
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Tidskriftsartikel (refereegranskat)abstract
- We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. © 2012 Springer Science + Business Media B.V.
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