1. IMP3 1 RITM – code Computation with stiff transport models presented by D.Kalupin 12th Meeting of the ITPA Transport Physics (TP) Topical Group 7-10 May 2007, EPFL, Lausanne, Switzerland Institut für Plasmaphysik, Forschungszentrum Jülich GmbH, EURATOM Association, D-52425 Jülich, Germany EFDA Integrated Tokamak Modelling (ITM) Task Force

2. IMP3 2 -There is a number of stiff transport models (GLF23, RITM, Weiland) used in European codes (ASTRA, CRONOS, JETTO and RITM) - New transport code, which is under development by the EFDA ITM Task Force, and will be to a large extend assembled from existing codes, should be capable of working with stiff models - Thus, methods of reliable, stable operation with stiff models is one of current and urgent tasks for the ITM-TF - RITM code has a long time experience of operation with transport coefficients being strongly non-linear functions of of plasma parameter gradients

3. IMP3 3 Why stiff transport models cause problems? General form of transport equation: After time discretization: With stiff transport models D and are strongly non-linear functions of gradients of plasma parameters, terms and can lead to numerical instabilities

4. IMP3 4 Methods to avoid these numerical instabilities used in present codes - reduction of the time step usually, the time step is reduced down to 10 -5 -10 -4 s, or even 10 -6 s, which requires large simulation time - smoothing of profiles and/or transport coefficients a plenty of smoothing routines is developed and used in different codes, (caution: such procedure can smooth away important physics) - reformulation of transport equations integral form of transport equations does not contain the radial derivatives of transport coefficients - developments to solvers

5. IMP3 5 RITM approach to solve transport equations Standard form of transport equation: RITM integral form: New variables: New differential equation for N does not include derivatives of transport coefficients and any assumption about the function behavior in a grid sell, that improves the convergence

6. IMP3 6 Numerical approach for solving of differential equations in RITM Introducing new variable

7. IMP3 7 Numerical approach for solving of differential equations in RITM Tokar et.al, Computer Phys. Communications, 175 (2006) 30-35 Coefficients are determined from continuity of in knots and from boundary conditions This homogeneous equation has Known analytical solutions: This approach requires less iterations to get the steady state solution and allows to obtain solution with enough accuracy for larger time step RITM solver finite volume Time step :

8. IMP3 8 Smoothing routines … Diverse smoothing routines have been impemented in RITM 2. Smoothing by parabolic curve: 1. Fitting by smooth spline function which consists of three segments: !!! Should be used carefully, as it can smooth away the physics…

9. IMP3 9 RITM transport model CORE TRANSPORT EDGE TRANSPORT !!! Resulting transport coefficients are strongly non- linear functions of gradients of plasma parameters

10. IMP3 10 Transport simulations Using RITM approach, numerical instabilities due to very fast change (in space and time) of transport coefficients are completely avoided The time step in simulations can be increased up to 0.1 s and the space resolution can be up to 1000 radial points The total CPU time consumption is several times smaller than with the standard approach RITM run (201 radial points) ~ 30 min JETTO run (101 radial points) ~ 5 hours Predictive calculations of L-H transition in TEXTOR with RITM

11. IMP3 11 Summary RITM approach to solve transport equations allows for using of stiff transport models and avoid numerical instabilities due to very fast change of transport coefficients This method can be applied for calculations with sufficiently larger time step It is going to be one of the options for the core transport solver in ITM-TF