5 research outputs found
Shock Regularization with Smoothness-Increasing Accuracy-Conserving Dirac-Delta Polynomial Kernels
A smoothness-increasing accuracy conserving filtering approach to the
regularization of discontinuities is presented for single domain spectral
collocation approximations of hyperbolic conservation laws. The filter is based
on convolution of a polynomial kernel that approximates a delta-sequence. The
kernel combines a order smoothness with an arbitrary number of
zero moments. The zero moments ensure a order accurate approximation
of the delta-sequence to the delta function. Through exact quadrature the
projection error of the polynomial kernel on the spectral basis is ensured to
be less than the moment error. A number of test cases on the advection
equation, Burger's equation and Euler equations in 1D and 2D shown that the
filter regularizes discontinuities while preserving high-order resolutio
Uncertainty quantification in Eulerian-Lagrangian simulations of (point-)particle-laden flows with data-driven and empirical forcing models
An uncertainty quantification framework is developed for Eulerian-Lagrangian
models of particle-laden flows, where the fluid is modeled through a system of
partial differential equations in the Eulerian frame and inertial particles are
traced as points in the Lagrangian frame. The source of uncertainty in such
problems is the particle forcing, which is determined empirically or
computationally with high-fidelity methods (data-driven). The framework relies
on the averaging of the deterministic governing equations with the stochastic
forcing and allows for an estimation of the first and second moment of the
quantities of interest. Via comparison with Monte Carlo simulations, it is
demonstrated that the moment equations accurately predict the uncertainty for
problems whose Eulerian dynamics are either governed by the linear advection
equation or the compressible Euler equations. In areas of singular particle
interfaces and shock singularities significant uncertainty is generated. An
investigation into the effect of the numerical methods shows that
low-dissipative higher-order methods are necessary to capture numerical
singularities (shock discontinuities, singular source terms, particle
clustering) with low diffusion in the propagation of uncertainty
An explicit semi-Lagrangian, spectral method for solution of Lagrangian transport equations in Eulerian-Lagrangian formulations
An explicit high order semi-Lagrangian method is developed for solving
Lagrangian transport equations in Eulerian-Lagrangian formulations. To ensure a
semi-Lagrangian approximation that is consistent with an explicit Eulerian,
discontinuous spectral element method (DSEM) discretization used for the
Eulerian formulation, Lagrangian particles are seeded at Gauss quadrature
collocation nodes within an element. The particles are integrated explicitly in
time to obtain an advected polynomial solution at the advected Gauss quadrature
locations. This approximation is mapped back in a semi-Lagrangian fashion to
the Gauss quadrature points through a least squares fit using constraints for
element boundary values and optional constraints for mass and energy
preservation. An explicit time integration is used for the semi-Lagrangian
approximation that is consistent with the grid based DSEM solver, which ensures
that particles seeded at the Gauss quadrature points do not leave the element's
bounds. The method is hence local and parallel and facilitates the solution of
the Lagrangian formulation without the grid complexity, and parallelization
challenges of a particle solver in particle-mesh methods. Numerical tests with
one and two dimensional advection equation are carried out. The method
converges exponentially. The use of mass and energy constraints can improve
accuracy depending on the accuracy of the time integration.Comment: 20 pages, 11 figure
Multi-element SIAC filter for shock capturing applied to high-order discontinuous Galerkin spectral element methods
We build a multi-element variant of the smoothness increasing accuracy
conserving (SIAC) shock capturing technique proposed for single element
spectral methods by Wissink et al. (B.W. Wissink, G.B. Jacobs, J.K. Ryan, W.S.
Don, and E.T.A. van der Weide. Shock regularization with smoothness-increasing
accuracy-conserving Dirac-delta polynomial kernels. Journal of Scientific
Computing, 77:579--596, 2018). In particular, the baseline scheme of our method
is the nodal discontinuous Galerkin spectral element method (DGSEM) for
approximating the solution of systems of conservation laws. It is well known
that high-order methods generate spurious oscillations near discontinuities
which can develop in the solution for nonlinear problems, even when the initial
data is smooth. We propose a novel multi-element SIAC filtering technique
applied to the DGSEM as a shock capturing method. We design the SIAC filtering
such that the numerical scheme remains high-order accurate and that the shock
capturing is applied adaptively throughout the domain. The shock capturing
method is derived for general systems of conservation laws. We apply the novel
SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics (MHD)
equations to several standard test problems with a variety of boundary
conditions
A high-order semi-Lagrangian method for the consistent Monte-Carlo solution of stochastic Lagrangian drift-diffusion models coupled with Eulerian discontinuous spectral element method
The explicit semi-Lagrangian method method for solution of Lagrangian
transport equations as developed in [Natarajan and Jacobs, Computer and Fluids,
2020] is adopted for the solution of stochastic differential equations that is
consistent with Discontinuous Spectral Element Method (DSEM) approximations of
Eulerian conservation laws. The method extends the favorable properties of DSEM
that include its high-order accuracy, its local and boundary fitted properties
and its high performance on parallel platforms for the concurrent Monte-Carlo,
semi-Lagrangian and Eulerian solution of a class of time-dependent problems
that can be described by coupled Eulerian-Lagrangian formulations. The
semi-Lagrangian method seeds particles at Gauss quadrature collocation nodes
within a spectral element. The particles are integrated explicitly in time
according to a drift velocity and a Wiener increment forcing and form the nodal
basis for an advected interpolant. This interpolant is mapped back in a
semi-Lagrangian fashion to the Gauss quadrature points through a least squares
fit using constraints for element boundary values. Stochastic Monte-Carlo
samples are averaged element-wise on the quadrature nodes. The stable explicit
time step Wiener increment is sufficiently small to prevent particles from
leaving the element's bounds. The semi-Lagrangian method is hence local and
parallel and does not have the grid complexity, and parallelization challenges
of the commonly used Lagrangian particle solvers in particle-mesh methods for
solution of Eulerian-Lagrangian formulations. Formal proof is presented that
the semi-Lagrangian algorithm evolves the solution according to the Eulerian
Fokker-Planck equation. Numerical tests in one and two dimensions for
drift-diffusion problems show that the method converges exponentially for
constant and non-constant advection and diffusion velocities