| 1. |
- Andersson, Anders
(författare)
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A modified Schwarz–Christoffel mapping for regions with piecewise smooth boundaries
- 2008
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier, Amsterdam. - 0377-0427 .- 1879-1778. ; 213:1, s. 56-70
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Tidskriftsartikel (refereegranskat)abstract
- A method where polygon corners in Schwarz-Christoffel mappings are rounded, is used to construct mappings from the upper half-plane to regions bounded by arbitrary piecewise smooth curves. From a given curve, a polygon is constructed by taking tangents to the curve in a number of carefully chosen so called tangent points. The Schwarz-Christoffel mapping for that polygon is then constructed and modified to round the corners.Since such a modification causes effects on the polygon outside the rounded corners, the parameters in the mapping have to be re-determined. This is done by comparing side-lengths in tangent polygons to the given curve and the curve produced by the modified Schwarz-Christoffel mapping. The set of equations that this comparison gives, can normally be solved using a quasi--Newton method.The resulting function maps the upper half--plane on a region bounded by a curve that apart from possible vertices is smooth, i.e., one time continuously differentiable, that passes through the tangent points on the given curve, has the same direction as the given curve in these points and changes direction monotonically between them. Furthermore, where the original curve has a vertex, the constructed curve has a vertex with the same inner angle.The method is especially useful for unbounded regions with smooth boundary curves that pass infinity as straight lines, such as channels with parallel walls at the ends. These properties are kept in the region produced by the constructed mapping.
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| 2. |
- Andersson, Anders, 1957-
(författare)
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Modified Schwarz–Christoffel mappings using approximate curve factors
- 2009
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 233:4, s. 1117-1127
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Tidskriftsartikel (refereegranskat)abstract
- The Schwarz–Christoffel mapping from the upper half-plane to a polygonal region in the complex plane is an integral of a product with several factors, where each factor corresponds to a certain vertex in the polygon. Different modifications of the Schwarz–Christoffel mapping in which factors are replaced with the so-called curve factors to achieve polygons with rounded corners are known since long times. Among other requisites, the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis. In this paper, the term approximate curve factor is defined such that many of the already known curve factors are included as special cases. Additionally, by alleviating the requisite on the argument from exact to asymptotic equality, new types of curve factors are introduced. While traditional curve factors have a C1 regularity, C∞ regular approximate curve factors can be constructed, resulting in smooth boundary curves when used in conformal mappings. Applications include modelling of wave scattering in waveguides. When using approximate curve factors in modified Schwarz–Christoffel mappings, numerical conformal mappings can be constructed that preserve two important properties in the waveguides. First, the direction of the boundary curve can be well controlled, especially towards infinity, where the application requires two straight parallel walls. Second, a smooth (C∞) boundary curve can be achieved.
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| 3. |
- Arévalo, Carmen, et al.
(författare)
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Constant coefficient linear multistep methods with step density control
- 2007
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427. ; 205:2, s. 891-900
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Tidskriftsartikel (refereegranskat)abstract
- In linear multistep methods with variable step size, the method's coefficients are functions of the step size ratios. The coefficients therefore need to be recomputed on every step to retain the method's proper order of convergence. An alternative approach is to use step density control to make the method adaptive. If the step size sequence is smooth, the method can use constant coefficients without losing its order of convergence. The paper introduces this new adaptive technique and demonstrates its feasibility with a few test problems. The technique works in perfect agreement with theory for a given step density function. For practical use, however, the density must be generated with data computed from the numerical solution. We introduce a local error tracking controller, which automatically adapts the density to computed data, and demonstrate in computational experiments that the technique works well at least up to fourth-order methods. (c) 2006 Elsevier B.V. All rights reserved.
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| 4. |
- Atle, Andreas, et al.
(författare)
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On surface radiation conditions for high-frequency wave scattering
- 2007
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427 .- 1879-1778. ; 204:2, s. 306-316
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Tidskriftsartikel (refereegranskat)abstract
- A new approximation of the logarithmic derivative of the Hankel function is derived and applied to high-frequency wave scattering. We re-derive the on surface radiation condition (OSRC) approximations that are well suited for a Dirichlet boundary in acoustics. These correspond to the Engquist-Majda absorbing boundary conditions. Inverse OSRC approximations are derived and they are used for Neumann boundary conditions. We obtain an implicit OSRC condition, where we need to solve a tridiagonal system. The OSRC approximations are well suited for moderate wave numbers. The approximation of the logarithmic derivative is also used for deriving a generalized physical optics approximation, both for Dirichlet and Neumann boundary conditions. We have obtained similar approximations in electromagnetics, for a perfect electric conductor. Numerical computations are done for different objects in 2D and 3D and for different wave numbers. The improvement over the standard physical optics is verified.
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| 5. |
- Bennewitz, Christer
(författare)
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Approximation numbers = singular values
- 2007
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427. ; 208:1, s. 102-110
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Tidskriftsartikel (refereegranskat)abstract
- This paper introduces a generalisation of the notion of singular value for Hilbert space operators to more general Banach spaces. It is shown that for a simple integral operator of Hardy type the singular values are the eigenvalues of a non-linear Sturm-Liouville equation and coincide with the approximation numbers of the operator. Finally, asymptotic formulas for the singular numbers are deduced. (c) 2006 Published by Elsevier B.V.
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| 6. |
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| 7. |
- Burman, Erik, et al.
(författare)
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A unified stabilized method for Stokes’ and Darcy's equations
- 2007
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427 .- 1879-1778. ; 198:1, s. 35-51
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Tidskriftsartikel (refereegranskat)abstract
- We use the lowest possible approximation order, piecewise linear, continuous velocities and piecewise constant pressures to compute solutions to Stokes equation and Darcy's equation, applying an edge stabilization term to avoid locking. We prove that the formulation satisfies the discrete inf-sup condition, we prove an optimal a priori error estimate for both problems. The formulation is then extended to the coupled case using a Nitsche-type weak formulation allowing for different meshes in the two subdomains. Finally, we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach.
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| 8. |
- Edström, Per
(författare)
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Numerical Performance of Stability Enhancing and Speed Increasing Steps in Radiative Transfer Solution Methods
- 2009
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427 .- 1879-1778. ; 228:1, s. 104-114
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Tidskriftsartikel (refereegranskat)abstract
- Methods for solving the radiative transfer problem, which are crucial for a number of sectors of industry, involve several numerical challenges. DORT2002 is a fast and numerically stable solution method for this problem, and has been implemented in MATLAB. This paper studies the numerical performance of DORT2002, in terms of stability, speed and accuracy. Focus is on the effects of the steps that are needed to make the method numerically efficient, and that differ from a naive implementation. Performance tests show that the steps that are included to improve stability and speed of DORT2002 are very successful. Together they give an unconditionally stable solution method to a problem previously considered numerically intractable, and decrease computation time compared to a naive implementation with a factor 1 000-10 000 in typical cases and with a factor up to and beyond 10 000 000 in extreme cases. It is also shown that the speed increasing steps are not introduced at the cost of reduced accuracy. Further studies and developments, that can have a positive impact on computation time, are suggested.
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| 9. |
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| 10. |
- Foufas, Georgios, 1977, et al.
(författare)
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Valuing Asian options using the finite element method and duality techniques
- 2008
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427 .- 1879-1778. ; 222:1, s. 144-158
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Tidskriftsartikel (refereegranskat)abstract
- The main objective of this paper is to develop an adaptive finite element method for computation of the values, and different sensitivity measures, of the Asian option with both fixed and floating strike. The pricing is based on Black-Scholes PDE-model and a method developed by Vecer where the resulting PDEs are of parabolic type in one spatial dimension and can be applied to both continuous and discrete Asian options. We propose using an adaptive finite element method which is based on a posteriori estimates of the error in desired quantities, which we derive using duality techniques. The a posteriori error estimates are tested and verified, and are used to calculate optimal meshes for each type of option. The use of adapted meshes gives superior accuracy and performance with less degrees of freedom than using uniform meshes. The suggested adaptive finite element method is stable, gives fast and accurate results, and can be applied to other types of options as well. (C) 2007 Elsevier B.V. All rights reserved.
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