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A generalized weighted residual method for RFP plasma simulation

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Jan Scheffel

Fusion Plasma Physics

Alfvén Laboratory, KTH

Stockholm, Sweden

• What is the GWRM?

• ODE example

• SIR - a globally convergent root solver

• Accuracy

• Efficiency

• Discussion

• Conclusion and prospects

Time differencing numerical initial value schemes (even implicit)

require extremelymany time steps for problems of physical

interest, where there are several separated time scales.

Causality is already embedded in the governing PDE’s –

- there is no need to mimic causality by time stepping.

Spectral methods (solution expanded in basis functions) are

successful in the spatial domain – why not employ them also in

the time domain?

By expanding in time + physical space + physical parameters, the

computational result will be semi-analytical. (Analytic basis

functions with numerical coefficients).

Ideal for scaling studies, for example.

What is the

Generalized Weighted Spectral Method (GWRM) ?

Fully spectral weighted residual method for semi-analytical solution of initial value partial differential equations.

- All time, spatial and physical parameter domains are represented by Chebyshev series, enabling closed and approximate analytical solutions. The method generalises earlier spatially spectral, finite time difference methods.
The method is acausal and thus avoids time step limitations.

The spectral coefficients are determined by iterative solution of a nonlinear system of algebraic eqs, for which a globally convergent semi-implicit root solver (SIR) has been developed.

- Accuracy is controlled by the number of included Chebyshev modes.
- Efficiency is controlled also by the use of temporal and spatial subdomains.
- Intended for efficient solution of nonlinear initial value problems in fluid mechanics and magnetohydrodynamics, including simulation of multi-time-scale RFP confinement and transport.

The Generalized Weighted Spectral Method (GWRM)

Consider a system of parabolic or hyperbolic initial-value PDE’s, symbolically written as

D is a nonlinear matrix operator, f is a forcing term.

D and f contains both physical variables and physical free parameters (denoted p).

Initial u(0,x;p) + (Dirichlet, Neumann or Robin) boundary u(t,xB;p) conditions.

Integrate in time:

Solution u(t,x;p) is approximated by finite, first kind Chebyshev polynomial series.

Definition: Chebyshev polynomial Tn(x) = cos(n arccosx).

For simplicity – here single equation, one spatial dimension x, one physical parameter p.

The Generalized Weighted Spectral Method (GWRM)

The Weighted Residual of the GWRM is given by

with

The TP-WRM coefficients are now obtained from the nonlinear system of algebraic equations

where

The initial state is expanded as

The Generalized Weighted Spectral Method (GWRM)

- COMMENTS
- • Boundary conditions are transformed into Chebyshev space (using Chebyshev interpolation);
- they enter at the highest modal numbers of the spatial Chebyshev coefficients.
- • All computations are in Chebyshev space.
- • Efficient procedures for integration, differentiation and nonlinear products in Chebyshev
- space have been developed.
- • Chebyshev polynomial expansions have several desirable qualities:
- - converge rapidly to the approximated function
- are real and can be converted to ordinary polynomials and vice versa
- minimax property - they are the most economical polynomial representation
- can be used for non-periodic boundary conditions

Simple GWRM example - the linear diffusion equation

Solution to be determined:

The coefficients aqrs

are determined

by iterations, using

a root solver.

for 1≤ q ≤ K + 1

with

Boundary conditions enter here

• What is the GWRM?

• ODE example

• SIR - a globally convergent root solver

• Accuracy

• Efficiency

• Discussion

• Conclusion and prospects

Light a match – a model of flame propagation:

y – flame radius

green - exact solution

d = 0.05; # Chebyshev modes K = 20, # time domains Nt = 1, error = 0.01

- = 0.01
- # Chebyshev modes K = 8
- # time domains Nt = 10
- error = 0.01

- = 0.0001 – Stiff problem!
- # Chebyshev modes K = 5
- # time domains Nt = 100
- error = 0.1
- Adaptive grid should be used
- for improved accuracy.

• What is the GWRM?

• ODE example

• SIR - a globally convergent root solver

• Accuracy

• Efficiency

• Discussion

• Conclusion and prospects

The GWRM - well adapted for iterative methods for two reasons:

1) Basic Chebyshev coefficient equations are of the standard iterative form

2) Initial estimate of solution vector can be chosen sufficiently close to the solution

by reducing the solution time interval

Instead of using direct iteration,

the Semi-Implicit Root solver (SIR)

finds the roots to the equations

or, in matrix form

The system

has the same solutions as the original system, but contains free parameters in the form

of the components of the matrix A. The parameters can be chosen to control

the gradients of the hypersurfaces . Adjusting these parameters, global, quasi-monotonous

and superlinear convergence is attained. In SIR,

whereas

are finite and is controlled; it produces limited step lengths, quasi-monotonous convergence;

and approaches zero after some initial iterations.

Newton’s method is a special case of the present method, when all

• Rapid second order convergence is generally approached after some iteration steps.

• Relationship to Newton’s method - approximately similar numerical work;

inversion of a Jacobian matrix at each iteration step.

Newton’s method

2D example

solution

Newton’s method with linesearch

2D example

local minimum

f ≠ 0

SIR

2D example

finds solution

• What is the GWRM?

• ODE example

• SIR - a globally convergent root solver

• Accuracy

• Efficiency

• Discussion

• Conclusion and prospects

- Burger’s nonlinear equation (parabolic)

Parameters:

Solution compared to Lax-Wendroff (explicit time differencing):

GWRM parameters - (S,M,N) = (2,7,6), 13 iterations.

L-W marginally stable parameters -

10-3

TP-WRM solution, using

two spatial subdomains

TP-WRM solution error, as

compared to exact solution

Result: GWRM 50 % faster than L-W

for same accuracy.

GWRM Burger equation solution,

including viscosity dependence u = u(t,x;v)

Initial conditionj(x) = x(1 - x) and boundary condition u(t,0;v) = u(t,1;v) = 0.

Solution shown versus x and v at time t = 2.5. Here K = 8, L = 10, and M = 2.

• What is the GWRM?

• ODE example

• SIR - a globally convergent root solver

• Accuracy

• Efficiency

• Discussion

• Conclusion and prospects

Wave equation, forced (hyperbolic)

Parameters:

Exact solution

GWRM solution

(averages out

fast time scale)

(slow + rapid time scale)

Efficiency

- Forced wave equation solutions u(t,x0) for fixed x = x0

GWRM

(K,L) = (6,8)

Lax-Wendroff, explicit

∆x = 1/30, 900 time steps

Crank-Nicholson, implicit

∆x = 1/30, 100 time steps

• What is the GWRM?

• ODE example

• SIR - a globally convergent root solver

• Accuracy

• Efficiency

• Discussion

• Conclusion and prospects

GWRM work so far:

• The time- and parameter-generalized weighted residual method,

J. Scheffel, 2008. (GWRM method outlined)

• Semi-analytical solution of initial-value problems,

D. Lundin, 2006. (Resistive MHD stability of RFP and z-pinch)

• Application of the time- and parameter generalized weighted reidual

method to systems of nonlinear equations,

D. Jackson, 2007. (Navier-Stokes equations, Rayleigh-Taylor instability)

• Further development and implementation of the GWRM

A. Mirza, ongoing Ph D studies (Application of GWRM to nonlinear resistive MHD)

SIR:

• Solution of systems of nonlinear equations, a semi-implicit approach,

J. Scheffel, 2006. (SIR outlined)

• Studies of a semi-implicit root solver,

C. Håkansson, M Sc Thesis. (Efficient SIR compared to other methods)

• The GWRM is shown to be accurate for spatially smooth solutions - convergence including sharp gradients should be further studied.

• Efficiency is central; SIR involves Jacobian matrix inversion of Chebyshev coefficient eqs - N eqs takes O[N3] operations.

• Methods to improve efficiency have been developed - temporal subdomains and spatial subdomain techniques using overlapping domains.

• Further benchmarking of efficiency for MHD relevant test problems should be carried out as well as corresponding comparisons with implicit methods.

• SIR efficiency and linking to GWRM is presently being optimized.

• What is the GWRM?

• ODE example

• SIR - a globally convergent root solver

• Accuracy

• Efficiency

• Discussion

• Conclusion and prospects

- A fully spectral method, the generalized weighted residual method (GWRM),
for solution of initial value partial differential equations, has been outlined.

- By representing all time, spatial and physical parameter domains
by Chebyshev series, semi-analytical solutions can be obtained as

ordinary polynomials.

(“Semi-analytical”: expansion in basis functions with numerical coefficients.)

Computed solutions thus contain r- t- and parameter dependence explicitly.

The method is global and avoids time step limitations.

- Spectral coefficients are found by iterative solution of a linear or nonlinear
system of algebraic equations, for which an efficient semi-implicit root solver

(SIR) has been developed.

- Accuracy is explicitly controlled by the number of modes and subdomains used.
To improve efficiency, a spatial subdomain approach has been developed.

Problems in fluid mechanics and MHD will be addressed.

- Future applications involve studies of nonlinear plasma instabilities at finite
plasma pressure in stochastic magnetic field geometries, in particular

operational limits in reversed-field pinches.