Statistical analysis and simulation of random shocks in stochastic Burgers equation

Abstract

We study the statistical properties of random shock waves in stochastic Burgers equation subject to random space-time perturbations and random initial conditions. By using the response-excitation probability density function (PDF) method and the Mori-Zwanzig (MZ) formulation of irreversible statistical mechanics, we derive exact reducedorder equations for the one-point and two-point PDFs of the solution field. In particular, we compute the statistical properties of random shock waves in the inviscid limit by using an adaptive (shock-capturing) discontinuous Galerkin method in both physical and probability spaces. We consider stochastic flows generated by high-dimensional random initial conditions and random additive forcing terms, yielding multiple interacting shock waves collapsing into clusters and settling down to a similarity state. We also address the question of how random shock waves in space and time manifest themselves in probability space. The proposed new mathematical framework can be applied to different conservation laws, potentially leading to new insights into highdimensional stochastic dynamical systems and more efficient computational algorithms.

Stochastic-deterministic modeling of bed load transport in shallow water flow over erodible slope: Linear stability analysis and numerical simulation

“…The major interest of this benchmark is to check whether the WENO method reproduces the non-local effect induced by the convective operator in the presence of stochastic forcing, which is equivalent to increasing the particle diffusivity as outlined below. Recent numerical studies on this topic focused on finite differences, adaptive Discontinuous Galerkin method and Total Variation Diminishing finite volume methods for the stochastic Burgers equation equation [7,24,41] but little is known about the performance of the WENO scheme for the continuous version of the stochastic Langevin equation with advection. Before we start with the numerical simulations, two caveats are in order.…”

The numerical approximation of nonlinear functionals and functional differential equations

“…converges to the FDE (95) uniformly in θ as m goes to infinity 13 . Next, we show that the solution to (99) is stable, i.e., that it is possible to bound its norm by a constant multiple of a suitable norm of the initial condition, and all the norms involved, do not depend on m. To this end, it is convenient to define…”

“…Hence, if the L ∞ norm of f 0 is bounded by a constant κ that is independent of m, then the problem (101) is stable in the L ∞ (R m ) norm 14 . Such strong bound implies that the solution 13 As we shall see in section 10, if the initial condition functional F0([θ]) associated with (95) is continuously Fréchet differentiable, then the solution to (95) is Fréchet differentiable as well. In this case, the residual defined in (97) is Fréchet differentiable.…”

“…Methods addressing the so-called uncertainty quantification (UQ) problem aim at representing such functional dependence based on polynomial chaos expansions[21,74,72,73], probabilistic collocation methods[23,19,77], or deep neural networks[53,79]. Other techniques rely on a reformulation of the problem in terms of kinetic equations[71,14] or hierarchies of kinetic equations[9,70,13].…”

Lattice Boltzmann model for predicting the deposition of inertial particles transported by a turbulent flow

“…According to this procedure, one introduces a 1D family of Gauss-Hermite Lattice Boltzmann models, denoted as GHLBM(N ; Q), where N is the order of the model and Q ≥ N + 1 is the quadrature order, i.e., the number of the vectors in the velocity set. The approximation of a PDF by using an expansion with respect to a family of orthogonal polynomials, followed by a quadrature method, is a quiet standard technique used in LB models [35,36,45,8,37]. Such a 1D…”