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Lindvall, K. & Scheffel, J. (2018). A time-spectral method for initial-value problems using a novel spatial subdomain scheme. COGENT MATHEMATICS, 5(1), Article ID 1529280.
Open this publication in new window or tab >>A time-spectral method for initial-value problems using a novel spatial subdomain scheme
2018 (English)In: COGENT MATHEMATICS, ISSN 2331-1835, Vol. 5, no 1, article id 1529280Article in journal (Refereed) Published
Abstract [en]

We analyse a novel subdomain scheme for time-spectral solution of initial-value partial differential equations. In numerical modelling spectral methods are commonplace for spatially dependent systems, whereas finite difference schemes are typically applied for the temporal domain. The Generalized Weighted Residual Method (GWRM) is a fully spectral method that spectrally decomposes all specified domains, including the temporal domain, using multivariate Chebyshev polynomials. The Common Boundary-Condition method (CBC) here proposed is a spatial subdomain scheme for the GWRM. It solves the physical equations independently from the global connection of subdomains in order to reduce the total number of modes. Thus, it is a condensation procedure in the spatial domain that allows for a simultaneous global temporal solution. It is here evaluated against the finite difference methods of Crank-Nicolson and Lax-Wendroff for two example linear PDEs. The CBC-GWRM is also applied to the linearised ideal magnetohydrodynamic (MHD) equations for a screw pinch equilibrium. The growth rate of the most unstable mode was efficiently computed with an error <0.1%.

Place, publisher, year, edition, pages
TAYLOR & FRANCIS AS, 2018
Keywords
time, spectral, weighted residual methods, MHD, Chebyshev, subdomain, PDE, initial, boundary
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-240022 (URN)10.1080/25742558.2018.1529280 (DOI)000451205400001 ()
Note

QC 20181210

Available from: 2018-12-10 Created: 2018-12-10 Last updated: 2018-12-10Bibliographically approved
Scheffel, J. & Lindvall, K. (2018). SIR—An efficient solver for systems of equations. Software Quality Professional, 7, 59-62
Open this publication in new window or tab >>SIR—An efficient solver for systems of equations
2018 (English)In: Software Quality Professional, ISSN 1522-0540, Vol. 7, p. 59-62Article in journal (Refereed) Published
Abstract [en]

The Semi-Implicit Root solver (SIR) is an iterative method for globally convergent solution of systems of nonlinear equations. We here present MATLAB and MAPLE codes for SIR, that can be easily implemented in any application where linear or nonlinear systems of equations need be solved efficiently. The codes employ recently developed efficient sparse matrix algorithms and improved numerical differentiation. SIR convergence is quasi-monotonous and approaches second order in the proximity of the real roots. Global convergence is usually superior to that of Newton's method, being a special case of the method. Furthermore the algorithm cannot land on local minima, as may be the case for Newton's method with line search. 

Place, publisher, year, edition, pages
Elsevier B.V., 2018
Keywords
Equation solver, MATLAB, Newton method, Root solver
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-227495 (URN)10.1016/j.softx.2018.01.003 (DOI)000457139300012 ()2-s2.0-85042934996 (Scopus ID)
Note

Export Date: 9 May 2018; Article; Correspondence Address: Lindvall, K.; Department of Fusion Plasma Physics, School of Electrical Engineering and Computer Science, KTH Royal Institute of TechnologySweden; email: kfli@kth.se. QC 20180516

Available from: 2018-05-16 Created: 2018-05-16 Last updated: 2019-02-26Bibliographically approved
Scheffel, J. (2014). Can time-spectral methods improve turbulence modelling?. In: 56th Annual Meeting of the APS Division of Plasma Physics, New Orleans, Louisiana, USA 27-31 October 2014: . Paper presented at 56th Annual Meeting of the APS Division of Plasma Physics, New Orleans, Louisiana, USA 27-31 October 2014.
Open this publication in new window or tab >>Can time-spectral methods improve turbulence modelling?
2014 (English)In: 56th Annual Meeting of the APS Division of Plasma Physics, New Orleans, Louisiana, USA 27-31 October 2014, 2014Conference paper, Poster (with or without abstract) (Refereed)
Abstract [en]

In computational fusion physics, the widely separated time and space scales often demand extremely long computer simulations and vast memory resources, using finite time steps. Gyrokinetic turbulence modelling at high Reynolds or Lundquist numbers may be allocated millions of CPU hours for parallel processing on supercomputers. It is thus worthwhile to explore new avenues that may alleviate requirements on computer power. Indeed, time-stepping may be completely avoided for initial-value problems. In the recently developed Generalized Weighted Residual Method GWRM [1], temporal, spatial and parameter domains are all handled using a solution ansatz in the form of a sum of Chebyshev polynomials. The coefficients of the ansatz are determined using a weighted residual method for which a new efficient equation solver has been developed [2]. In addition, the temporal and spatial computational region has been successfully treated using subdomain methods in a number of test problems, more efficiently than relevant finite difference methods. The GWRM, however, relies on solution of linear systems of equations in each subdomain, and memory requirement is an issue. In this presentation we will discuss recent subdomain approaches for efficient and convergent modelling of drift-wave turbulence.   

[1] Scheffel J, Partial Differential Equations: Theory, Analysis and Applications (Nova Science Publishers) p 1-49, 2011.

[2] Scheffel J and Håkansson C, Appl. Numer. Math. 59(2009)2430.

Keywords
Time-spectral, pde, weighted residual method, initial-value problem, turbulence, computational methods
National Category
Physical Sciences Fusion, Plasma and Space Physics
Research subject
Physics
Identifiers
urn:nbn:se:kth:diva-186432 (URN)
Conference
56th Annual Meeting of the APS Division of Plasma Physics, New Orleans, Louisiana, USA 27-31 October 2014
Note

QC 20160512

Available from: 2016-05-11 Created: 2016-05-11 Last updated: 2019-02-12Bibliographically approved
Scheffel, J., Schnack, D. D. & MIrza, A. A. (2013). Static current profile control and RFP confinement. Nuclear Fusion, 53(11), 113007
Open this publication in new window or tab >>Static current profile control and RFP confinement
2013 (English)In: Nuclear Fusion, ISSN 0029-5515, E-ISSN 1741-4326, Vol. 53, no 11, p. 113007-Article in journal (Refereed) Published
Abstract [en]

Static current profile control (CPC) is shown numerically to substantially enhance plasma confinement in the reversed-field pinch (RFP). By suitable application of an auxiliary electric field and adjustment of its internal location, width and amplitude, strongly decreased levels of dynamo fluctuations are obtained. The simulations are performed using a fully non-linear, resistive magnetohydrodynamic model, including the effects of ohmic heating as well as parallel and perpendicular heat conduction along stochastic field lines. The importance of controlling the parallel current profile in the core plasma to minimize the effects of tearing modes on confinement is thus confirmed. A near three-fold increase in energy confinement is found and poloidal plasma beta increases by 30% from 0.20 to 0.27. The edge heat flux is reduced to a third of that of the conventional RFP. The high-confinement phase is interrupted here by a crash, characterized by a rapid decrease in confinement. A detailed study of the crash phase is carried out by the standard Delta' theory and a fully resistive linearized time-spectral method; the generalized weighted residual method. The analysis suggests that the instability is caused by pressure-driven, resistive g-modes. Inclusion of anisotropic thermal conduction reduces the linear growth rates. As compared with our earlier numerical studies of CPC in the RFP, employing feedback control, the present static control scheme should be more easily implemented experimentally.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2013
Keywords
Reversed-Field Pinch, Current Drive, High-Beta, Fluctuation, Reduction
National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-121596 (URN)10.1088/0029-5515/53/11/113007 (DOI)000326684400008 ()2-s2.0-84887107572 (Scopus ID)
Note

QC 20131209

Available from: 2013-05-02 Created: 2013-05-02 Last updated: 2017-12-06Bibliographically approved
Scheffel, J. (2012). A spectral method in time for initial-value problems. American Journal of Computational Mathematics, 2(3), 173-193
Open this publication in new window or tab >>A spectral method in time for initial-value problems
2012 (English)In: American Journal of Computational Mathematics, ISSN 2161-1211, Vol. 2, no 3, p. 173-193Article in journal (Refereed) Published
Abstract [en]

A time-spectral method for solution of initial value partial differential equations is outlined. Multivariate Chebyshev series are used to represent all temporal, spatial and physical parameter domains in this generalized weighted residual method (GWRM). The approximate solutions obtained are thus analytical, finite order multivariate polynomials. The method avoids time step limitations. To determine the spectral coefficients, a system of algebraic equations is solved iteratively. A root solver, with excellent global convergence properties, has been developed. Accuracy and efficiency are controlled by the number of included Chebyshev modes and by use of temporal and spatial subdomains. As examples of advanced application, stability problems within ideal and resistive magnetohydrodynamics (MHD) are solved. To introduce the method, solutions to a stiff ordinary differential equation are demonstrated and discussed. Subsequently, the GWRM is applied to the Burger and forced wave equations. Comparisons with the explicit Lax-Wendroff and implicit Crank-Nicolson finite difference methods show that the method is accurate and efficient. Thus the method shows potential for advanced initial value problems in fluid mechanics and MHD.

Place, publisher, year, edition, pages
Scientific Research Publishing, 2012
Keywords
initial-value problem, WRM, time-spectral, spectral method, Chebyshev polynomial, fluid mechanics, MHD
National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-91442 (URN)10.4236/ajcm.2012.23023 (DOI)
Note

QC 20121127

Available from: 2012-11-27 Created: 2012-03-15 Last updated: 2012-11-27Bibliographically approved
MIrza, A. A., Scheffel, J. & Johnson, T. (2012). Effect of thermal conduction on pressure-driven modes in the reversed-field pinch. Nuclear Fusion, 52(12), 123012
Open this publication in new window or tab >>Effect of thermal conduction on pressure-driven modes in the reversed-field pinch
2012 (English)In: Nuclear Fusion, ISSN 0029-5515, E-ISSN 1741-4326, Vol. 52, no 12, p. 123012-Article in journal (Refereed) Published
Abstract [en]

Classical linearized resistive magnetohydrodynamic (MHD) stability theory predicts unstable pressure-driven modes even at low plasma beta values for the reversed-field pinch (RFP) because of its unfavourable curvature and strong poloidal magnetic field. These resistive g-modes undermine energy confinement and are detrimental to the RFP reactor potential. In the analysis, one aspect is common, which is the usage of the adiabatic energy equation, ignoring the contribution due to thermal conduction effects. However, in recent analysis, stabilization of pressure-driven modes is demonstrated through inclusion of thermal conductivity. In this paper, we compare the results obtained from both classical and thermal conduction modified boundary layer stability analysis with those from a time-spectral resistive linearized MHD code. Ohmic heating and thermal conduction effects are included in the calculations. We have found that thermal conduction effects stabilize pressure-driven resistive g-modes only for very low values of plasma beta. In addition, analytical and numerical investigation of the equilibrium reveal that, for reactor relevant values of S-0 and tearing stable plasmas, the scaling gamma similar to S-0(-1/5) for the growth rate of these modes is weaker than that for the adiabatic case gamma similar to S-0(-1/3).

Keywords
Magnetohydrodynamic Transport Model, Resistive Instabilities, Toroidal Plasma, Configurations, Systems
National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-105494 (URN)10.1088/0029-5515/52/12/123012 (DOI)000311754900015 ()2-s2.0-84870162299 (Scopus ID)
Note

QC 20130109

Available from: 2012-11-21 Created: 2012-11-21 Last updated: 2017-12-07Bibliographically approved
Scheffel, J. & MIrza, A. A. (2012). Resistive pressure driven RFP modes are not removed by heat conduction effects. In: 39th EPS Conference on Plasma Physics 2012, EPS 2012 and the 16th International Congress on Plasma Physics: Volume 3, 2012. Paper presented at 39th EPS Conference on Plasma Physics and 16th Int. Congress on Plasma Physics, Stockholm, Sweden, 2-6 July 2012 (pp. 1690-1693).
Open this publication in new window or tab >>Resistive pressure driven RFP modes are not removed by heat conduction effects
2012 (English)In: 39th EPS Conference on Plasma Physics 2012, EPS 2012 and the 16th International Congress on Plasma Physics: Volume 3, 2012, 2012, p. 1690-1693Conference paper, Published paper (Refereed)
Abstract [en]

During the last decade it has been shown theoretically, numerically and experimentally that current driven, resistive tearing modes can be significantly suppressed in the reversed-field pinch (RFP). In these advanced scenarios, the confinement time can be enhanced by a factor 5-10. Pressure driven resistive instabilities (g-modes) still stand in the way, however, for high RFP confinement. Classical theory [1] shows that the unfavourable RFP curvature inevitably leads to unacceptably large linear growth rates even at high Lundquist numbers. Later theory [2] demonstrates, however, that the classical assumption of adiabatic plasma energy dynamics is inaccurate. The reason is that anomalously large experimental perpendicular heat conduction, together with strong parallel heat conduction, to a certain extent outbalance the pressure terms of the plasma energy equation. Resulting resistive length scales appear to extend the resistive layer at the resonance to allow for fully stable, finite beta RFP configurations. In the present work we show theoretically that the latter result is limited to low beta only and that it scales unfavourably with Lundquist number. Numerical solution, using a novel time-spectral method [3] of the linearised resistive MHD initial-value equations including heat conduction, ohmic heating and resistivity, supports the analytical results

National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-105504 (URN)2-s2.0-84876917604 (Scopus ID)978-162276981-0 (ISBN)
Conference
39th EPS Conference on Plasma Physics and 16th Int. Congress on Plasma Physics, Stockholm, Sweden, 2-6 July 2012
Note

QC 20130115

Available from: 2012-11-21 Created: 2012-11-21 Last updated: 2013-09-10Bibliographically approved
Scheffel, J. (2012). Time-spectral solution of initial-value problems. In: 54th Annual Meeting of the APS Division of Plasma Physics; Providence, Rhode Island, USA, 29 October – 2 November 2012: . Paper presented at 54th Annual Meeting of the APS Division of Plasma Physics; Providence, Rhode Island, USA, 29 October – 2 November 2012.
Open this publication in new window or tab >>Time-spectral solution of initial-value problems
2012 (English)In: 54th Annual Meeting of the APS Division of Plasma Physics; Providence, Rhode Island, USA, 29 October – 2 November 2012, 2012Conference paper, Oral presentation with published abstract (Refereed)
Abstract [en]

A time-spectral method for solutions of initial-value partial differential equations has recently been developed [1]. The purpose of the method is to avoid inefficient time stepping for problems in plasma physics with widely separated time scales. Temporal, spatial and parameter domains are all treated using an ansatz in the form of a sum of Chebyshev polynomials. The coefficients of the ansatz is determined using a generalized weighted residual method. A new, efficient solver for the resulting algebraic systems of coefficient equations has been developed [2]. In addition, subdomain methods for the temporal and spatial domains are employed [3]. The question is now: to what extent are time-spectral methods really more attractive than finite difference methods? We will report on results concerning accuracy and efficiency for several linear and nonlinear model partial differential equations.

National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-129340 (URN)
Conference
54th Annual Meeting of the APS Division of Plasma Physics; Providence, Rhode Island, USA, 29 October – 2 November 2012
Note

QC 20130930

Available from: 2013-09-27 Created: 2013-09-27 Last updated: 2019-02-12Bibliographically approved
Scheffel, J. & Mirza, A. A. (2012). Time-spectral solution of initial-value problems – subdomain approach. American Journal of Computational Mathematics, 2(2), 72-81
Open this publication in new window or tab >>Time-spectral solution of initial-value problems – subdomain approach
2012 (English)In: American Journal of Computational Mathematics, ISSN 2161-1211, Vol. 2, no 2, p. 72-81Article in journal (Refereed) Published
Abstract [en]

Temporal and spatial subdomain techniques are proposed for a time-spectral method for solution of initial-value problems. The spectral method, called the generalized weighted residual method (GWRM), is a generalization of weighted residual methods to the time and parameter domains [1]. A semi-analytical Chebyshev polynomial ansatz is employed, and the problem reduces to determine the coefficients of the ansatz from linear or nonlinear algebraic systems of equations. In order to avoid large memory storage and computational cost, it is preferable to subdivide the temporal and spatial domains into subdomains. Methods and examples of this article demonstrate how this can be achieved. 

Place, publisher, year, edition, pages
Scientific Research Publishing, 2012
Keywords
initial-value problem, time-spectral, spectral method, subdomains, domain decomposition
National Category
Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-91443 (URN)10.4236/ajcm.2012.22010 (DOI)
Note

QC 20121127

Available from: 2012-11-27 Created: 2012-03-15 Last updated: 2013-05-02Bibliographically approved
Mirza, A. A. & Scheffel, J. (2011). Numerical study of thermal conductivity effects on stability of the reversed-field pinch. In: 38th EPS Conference on Plasma Physics, Strasbourg, France, 27 June – 1 July 2011: . Paper presented at 38th EPS Conference on Plasma Physics, Strasbourg, France, 27 June – 1 July 2011.
Open this publication in new window or tab >>Numerical study of thermal conductivity effects on stability of the reversed-field pinch
2011 (English)In: 38th EPS Conference on Plasma Physics, Strasbourg, France, 27 June – 1 July 2011, 2011Conference paper, Published paper (Refereed)
National Category
Natural Sciences Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-76038 (URN)2-s2.0-84867632865 (Scopus ID)
Conference
38th EPS Conference on Plasma Physics, Strasbourg, France, 27 June – 1 July 2011
Note

QC 20120413

Available from: 2012-02-06 Created: 2012-02-06 Last updated: 2019-01-16Bibliographically approved
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