82 research outputs found
Parallel-in-time integration of the shallow water equations on the rotating sphere using Parareal and MGRIT
Despite the growing interest in parallel-in-time methods as an approach to
accelerate numerical simulations in atmospheric modelling, improving their
stability and convergence remains a substantial challenge for their application
to operational models. In this work, we study the temporal parallelization of
the shallow water equations on the rotating sphere combined with time-stepping
schemes commonly used in atmospheric modelling due to their stability
properties, namely an Eulerian implicit-explicit (IMEX) method and a
semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to
investigate the performance of parallel-in-time methods, namely Parareal and
Multigrid Reduction in Time (MGRIT), when these well-established schemes are
used on the coarse discretization levels and provide insights on how they can
be improved for better performance. We begin by performing an analytical
stability study of Parareal and MGRIT applied to a linearized ordinary
differential equation depending on the choice of a coarse scheme. Next, we
perform numerical simulations of two standard tests to evaluate the stability,
convergence and speedup provided by the parallel-in-time methods compared to a
fine reference solution computed serially. We also conduct a detailed
investigation on the influence of artificial viscosity and hyperviscosity
approaches, applied on the coarse discretization levels, on the performance of
the temporal parallelization. Both the analytical stability study and the
numerical simulations indicate a poorer stability behaviour when SL-SI-SETTLS
is used on the coarse levels, compared to the IMEX scheme. With the IMEX
scheme, a better trade-off between convergence, stability and speedup compared
to serial simulations can be obtained under proper parameters and artificial
viscosity choices, opening the perspective of the potential competitiveness for
realistic models.Comment: 35 pages, 23 figure
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Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)
This small collaborative workshop brought together
experts from the Sino-German project working in the field of advanced numerical methods for
hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the
convergence of numerical methods and proper solution concepts were addressed as well
Entropy-Preserving and Entropy-Stable Relaxation IMEX and Multirate Time-Stepping Methods
We propose entropy-preserving and entropy-stable partitioned Runge--Kutta
(RK) methods. In particular, we extend the explicit relaxation Runge--Kutta
methods to IMEX--RK methods and a class of explicit second-order multirate
methods for stiff problems arising from scale-separable or grid-induced
stiffness in a system. The proposed approaches not only mitigate system
stiffness but also fully support entropy-preserving and entropy-stability
properties at a discrete level. The key idea of the relaxation approach is to
adjust the step completion with a relaxation parameter so that the
time-adjusted solution satisfies the entropy condition at a discrete level. The
relaxation parameter is computed by solving a scalar nonlinear equation at each
timestep in general; however, as for a quadratic entropy function, we
theoretically derive the explicit form of the relaxation parameter and
numerically confirm that the relaxation parameter works the Burgers equation.
Several numerical results for ordinary differential equations and the Burgers
equation are presented to demonstrate the entropy-conserving/stable behavior of
these methods. We also compare the relaxation approach and the incremental
direction technique for the Burgers equation with and without a limiter in the
presence of shocks.Comment: 37 pages, 16 figures, 4 table
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Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws
The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling.
Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms
High Order Asymptotic Preserving and Classical Semi-implicit RK Schemes for the Euler-Poisson System in the Quasineutral Limit
In this paper, the design and analysis of high order accurate IMEX finite
volume schemes for the compressible Euler-Poisson (EP) equations in the
quasineutral limit is presented. As the quasineutral limit is singular for the
governing equations, the time discretisation is tantamount to achieving an
accurate numerical method. To this end, the EP system is viewed as a
differential algebraic equation system (DAEs) via the method of lines. As a
consequence of this vantage point, high order linearly semi-implicit (SI) time
discretisation are realised by employing a novel combination of the direct
approach used for implicit discretisation of DAEs and, two different classes of
IMEX-RK schemes: the additive and the multiplicative. For both the time
discretisation strategies, in order to account for rapid plasma oscillations in
quasineutral regimes, the nonlinear Euler fluxes are split into two different
combinations of stiff and non-stiff components. The high order scheme resulting
from the additive approach is designated as a classical scheme while the one
generated by the multiplicative approach possesses the asymptotic preserving
(AP) property. Time discretisations for the classical and the AP schemes are
performed by standard IMEX-RK and SI-IMEX-RK methods, respectively so that the
stiff terms are treated implicitly and the non-stiff ones explicitly. In order
to discretise in space a Rusanov-type central flux is used for the non-stiff
part, and simple central differencing for the stiff part. AP property is also
established for the space-time fully-discrete scheme obtained using the
multiplicative approach. Results of numerical experiments are presented, which
confirm that the high order schemes based on the SI-IMEX-RK time discretisation
achieve uniform second order convergence with respect to the Debye length and
are AP in the quasineutral limit
Monolithic Multigrid for Magnetohydrodynamics
The magnetohydrodynamics (MHD) equations model a wide range of plasma physics
applications and are characterized by a nonlinear system of partial
differential equations that strongly couples a charged fluid with the evolution
of electromagnetic fields. After discretization and linearization, the
resulting system of equations is generally difficult to solve due to the
coupling between variables, and the heterogeneous coefficients induced by the
linearization process. In this paper, we investigate multigrid preconditioners
for this system based on specialized relaxation schemes that properly address
the system structure and coupling. Three extensions of Vanka relaxation are
proposed and applied to problems with up to 170 million degrees of freedom and
fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to
20,000 for time-dependent problems
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