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%.

Transport phenomena in fusion plasma pose a daunting task for both real-time experiments and numerical modelling. The transport is driven by micro-instabilities caused by a host of unstable modes, for example ion temperature gradient and trapped electron modes. These modes can be modelled using fluid or gyrokinetic equations. However, the equations are characterised by high degrees of freedom and high temporal and spatial numerical requirements. Thus, a time-spectral method GWRM has been developed in order to efficiently solve these multiple time scale equations. The GWRM assumes a multivariate Chebyshev expansion ansatz in time, space, and parameter domain. Advantages are that time constraining CFL criteria no longer apply and that the solution accurately averages over small time-scale dynamics. For benchmarking, a two-fluid 2D drift wave turbulence model has been solved in order to study toroidal ion temperature gradient growth rates and nonlinear behaviour.

3.

Lindvall, Kristoffer

et al.

KTH, School of Electrical Engineering and Computer Science (EECS), Fusion Plasma Physics.

Scheffel, Jan

KTH, School of Electrical Engineering and Computer Science (EECS), Fusion Plasma Physics.

Numerical weather prediction (NWP) is a difficult task in chaotic dynamical regimes because of the strong sensitivity to initial conditions and physical parameters. As a result, high numerical accuracy is usually necessary. In this chapter, an accurate and efficient alternative to the traditional time stepping solution methods is presented; the time-spectral method. The generalized weighted residual method (GWRM) solves systems of nonlinear ODEs and PDEs using a spectral representation of time. Not being subject to CFL-like criteria, the GWRM typically employs time intervals two orders of magnitude larger than those of time-stepping methods. As an example, efficient solution of the chaotic Lorenz 1984 equations is demonstrated. The results indicate that the method has strong potential for NWP. Furthermore, employing spectral representations of physical parameters and initial values, families of solutions are obtained in a single computation. Thus, the GWRM is conveniently used for studies of system parameter dependency and initial condition error growth in NWP.

The Generalized Weighted Residual Method (GWRM) is a time-spectral method for solving initial value partial differential equations. The GWRM treats the temporal, spatial, and parameter domains by projecting the residual to a Chebyshev polynomial space, with the variational principle being that the residual is zero. This treatment provides a global semi-analytical solution. However, straightforward global solution is not economical. One remedy is the inclusion of spatial and temporal sub-domains with coupled internal boundary conditions, which decreases memory requirements and introduces sparse matrices. Only the equations pertaining to the boundary conditions need be solved globally, making the method parallelizable in time. Efficient solution of the linearized ideal MHD stability equations of screw-pinch equilibria are proved possible. The GWRM has also been used to solve strongly nonlinear ODEs such as the Lorenz equations (1984), and is capable of competing with finite time difference schemes in terms of both accuracy and efficiency. GWRM solutions of linear and nonlinear model problems of interest for stability and turbulence modelling will be presented, including detailed comparisons with time stepping methods.

5.

Scheffel, Jan

et al.

KTH, School of Electrical Engineering and Computer Science (EECS), Fusion Plasma Physics.

Lindvall, Kristoffer

KTH, School of Electrical Engineering and Computer Science (EECS), Fusion Plasma Physics.

Time-spectral solution of ordinary and partial differential equations is often regarded as an inefficient approach. The associated extension of the time domain, as compared to finite difference methods, is believed to result in uncomfortably many numerical operations and high memory requirements. It is shown in this work that performance is substantially enhanced by the introduction of algorithms for temporal and spatial subdomains in combination with sparse matrix methods. The accuracy and efficiency of the recently developed time spectral, generalized weighted residual method (GWRM) are compared to that of the explicit Lax-Wendroff and implicit Crank-Nicolson methods. Three initial-value PDEs are employed as model problems; the 1D Burger equation, a forced 1D wave equation and a coupled system of 14 linearized ideal magnetohydrodynamic (MHD) equations. It is found that the GWRM is more efficient than the time-stepping methods at high accuracies. The advantageous scalings Nt**1.0*Ns**1.43 and Nt**0.0*Ns**1.08 were obtained for CPU time and memory requirements, respectively, with Nt and Ns denoting the number of temporal and spatial subdomains. For time-averaged solution of the two-time-scales forced wave equation, GWRM performance exceeds that of the finite difference methods by an order of magnitude both in terms of CPU time and memory requirement. Favorable subdomain scaling is demonstrated for the MHD equations, indicating a potential for efficient solution of advanced initial-value problems in, for example, fluid mechanics and MHD.

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.

Finite difference methods are traditionally used for modelling the time domain in numerical weather prediction (NWP). Time-spectral solution is an attractive alternative for reasons of accuracy and efficiency and because time step limitations associated with causal CFL-like criteria, typical for explicit finite difference methods, are avoided. In this work, the Lorenz 1984 chaotic equations are solved using the time-spectral algorithm GWRM (Generalized Weighted Residual Method). Comparisons of accuracy and efficiency are carried out for both explicit and implicit time-stepping algorithms. It is found that the efficiency of the GWRM compares well with these methods, in particular at high accuracy. For perturbative scenarios, the GWRM was found to be as much as four times faster than the finite difference methods. A primary reason is that the GWRM time intervals typically are two orders of magnitude larger than those of the finite difference methods. The GWRM has the additional advantage to produce analytical solutions in the form of Chebyshev series expansions. The results are encouraging for pursuing further studies, including spatial dependence, of the relevance of time-spectral methods for NWP modelling. Program summary: Program Title: Time-adaptive GWRM Lorenz 1984 Program Files doi: http://dx.doi.org/10.17632/4nxfyjj7nv.1 Licensing provisions: MIT Programming language: Maple Nature of problem: Ordinary differential equations with varying degrees of complexity are routinely solved with numerical methods. The set of ODEs pertaining to chaotic systems, for instance those related to numerical weather prediction (NWP) models, are highly sensitive to initial conditions and unwanted errors. To accurately solve ODEs such as the Lorenz equations (E. N. Lorenz, Tellus A 36 (1984) 98–110), small time steps are required by traditional time-stepping methods, which can be a limiting factor regarding the efficiency, accuracy, and stability of the computations. Solution method: The Generalized Weighted Residual Method, being a time-spectral algorithm, seeks to increase the time intervals in the computation without degrading the efficiency, accuracy, and stability. It does this by postulating a solution ansatz as a sum of weighted Chebyshev polynomials, in combination with the Galerkin method, to create a set of linear/non-linear algebraic equations. These algebraic equations are then solved iteratively using a Semi Implicit Root solver (SIR), which has been chosen due to its enhanced global convergence properties. Furthermore, to achieve a desired accuracy across the entire domain, a time-adaptive algorithm has been developed. By evaluating the magnitudes of the Chebyshev coefficients in the time dimension of the solution ansatz, the time interval can either be decreased or increased.