Algorithm 731

Algorithm 731: A moving‐grid interface for systems of one‐dimensional time‐dependent partial differential equations. In the last decade, several numerical techniques have been developed to solve time-dependent partial differential equations (PDEs) in one dimension having solutions with steep gradients in space and in time. One of these techniques, a moving-grid method based on a Lagrangian description of the PDE and a smoothed-equidistribution principle to define the grid positions at each time level, has been coupled with a spatial discretization method that automatically discreizes the spatial part of the user-defined PDE following the method of lines approach. We supply two FORTRAN subroutines, CWRESU and CWRESX, which compute the residuals of the differential algebraic equations (DAE) system obtained from semidiscretizing, respectively, the PDE and the set of moving-grid equations. These routines are combined in an enveloping routine SKMRES, which delivers the residuals of the complete DAE system. To solve this stiff, nonlinear DAE system, a robust and efficient time-integrator must be applied, for example, a BDF method such as implemented in the DAE solvers SPRINT [Berzins and Furzeland 1985; 1986; Berzins et al. 1989] and DASSL [Brenan et al. 1989; Petzold 1983]. Some numerical examples are shown to illustrate the simple and effective use of this software interface.

This software is also peer reviewed by journal TOMS.

References in zbMATH (referenced in 34 articles )

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  1. Saramito, Pierre; Smutek, Claude; Cordonnier, Beno{^ı}t: Numerical modeling of shallow non-Newtonian flows. I: The 1D horizontal dam break problem revisited (2013)
  2. Balmforth, Neil J.; Forterre, Y.; Pouliquen, O.: The viscoplastic Stokes layer (2009)
  3. Chen, Wan; Ward, Michael J.: Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model (2009)
  4. Doelman, Arjen; van Heijster, Peter; Kaper, Tasso J.: Pulse dynamics in a three-component system: Existence analysis (2009)
  5. Saucez, P.; Some, L.; Vande Wouwer, A.: Matlab implementation of a moving grid method based on the equidistribution principle (2009)
  6. Wang, R.; Keast, P.; Muir, P.H.: Algorithm 874: BACOLR - spatial and temporal error control software for PDEs based on high-order adaptive collocation. (2008)
  7. Beck, M.; Doelman, A.; Kaper, T.J.: A geometric construction of traveling waves in a bioremediation model (2006)
  8. Craster, R.V.; Matar, O.K.: Electrically induced pattern formation in thin leaky dielectric films (2005)
  9. Dikansky, Arnold: Fitzhugh-Nagumo equations in a nonhomogeneous medium (2005)
  10. Vande Wouwer, A.; Saucez, P.; Schiesser, W.E.; Thompson, S.: A MATLAB implementation of upwind finite differences and adaptive grids in the method of lines (2005)
  11. Zegeling, P.A.; de Boer, W.D.; Tang, H.Z.: Robust and efficient adaptive moving mesh solution of the 2-D Euler equations (2005)
  12. Doelman, Arjen; Iron, David; Nishiura, Yasumasa: Destabilization of fronts in a class of bistable systems (2004)
  13. Morgan, David S.; Kaper, Tasso J.: Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots (2004)
  14. Wang, R.; Keast, P.; Muir, P.: BACOL: B-spline adaptive collocation software for 1-D parabolic PDEs (2004)
  15. Wang, R.; Keast, P.; Muir, P.: A high-order global spatially adaptive collocation method for 1-D parabolic PDEs (2004)
  16. Wang, Rong; Keast, Patrick; Muir, Paul: A comparison of adaptive software for 1D parabolic PDEs (2004)
  17. Cheng, H.; Lin, P.; Sheng, Q.; Tan, R. C. E.: Solving degenerate reaction-diffusion equations via variable step Peaceman-Rachford splitting (2003)
  18. Doelman, Arjen; Kaper, Tasso J.: Semistrong pulse interactions in a class of coupled reaction-diffusion equations (2003)
  19. Craster, R.V.; Matar, O.K.; Papageorgiou, D.T.: Pinchoff and satellite formation in surfactant covered viscous threads (2002)
  20. Mackenzie, J. A.; Robertson, M. L.: A moving mesh method for the solution of the one-dimensional phase-field equations (2002)

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