Implicit and semiimplicit schemes: algorithms
Keppens, R. and Toth, G. and Botchev, M.A. and Ploeg, A. van der (1999) Implicit and semiimplicit schemes: algorithms. International Journal for Numerical Methods in Fluids, 30 (3). pp. 335352. ISSN 02712091

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Abstract:  This study formulates general guidelines to extend an explicit code with a great variety of implicit and semiimplicit time integration schemes. The discussion is based on their specific implementation in the Versatile Advection Code, which is a general purpose software package for solving systems of nonlinear hyperbolic (and/or parabolic) partial differential equations, using standard high resolution shock capturing schemes. For all combinations of explicit high resolution schemes with implicit and semiimplicit treatments, it is shown how secondorder spatial and temporal accuracy for the smooth part of the solutions can be maintained. Strategies to obtain steady state and time accurate solutions implicitly are discussed. The implicit and semiimplicit schemes require the solution of large linear systems containing the Jacobian matrix. The Jacobian matrix itself is calculated numerically to ensure the generality of this implementation. Three options are discussed in terms of applicability, storage requirements and computational efficiency. One option is the easily implemented matrixfree approach, but the Jacobian matrix can also be calculated by using a general grid masking algorithm, or by an efficient implementation for a specific LaxFriedrichtype total variation diminishing (TVD) spatial discretization. The choice of the linear solver depends on the dimensionality of the problem. In one dimension, a direct block tridiagonal solver can be applied, while in more than one spatial dimension, a conjugate gradient (CG)type iterative solver is used. For advectiondominated problems, preconditioning is needed to accelerate the convergence of the iterative schemes. The modified block incomplete LUpreconditioner is implemented, which performs very well. Examples from twodimensional hydrodynamic and magnetohydrodynamic computations are given. They model transonic stellar outflow and recover the complex magnetohydrodynamic bow shock flow in the switchon regime found in De Sterck et al. [Phys. Plasmas, 5, 4015 (1998)]. 
Item Type:  Article 
Faculty:  Electrical Engineering, Mathematics and Computer Science (EEMCS) 
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Link to this item:  http://purl.utwente.nl/publications/66811 
Official URL:  https://doi.org/10.1002/(SICI)10970363(19990615)30:3<335::AIDFLD923>3.0.CO;2U 
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