A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations

Abstract

We consider the numerical solution of scalar, nonlinear degenerate convectiondiffusion problems with random diffusion coefficient and with random flux functions. Building on recent results on the existence, uniqueness and continuous dependence of weak solutions on data in the deterministic case, we develop a definition of random entropy solution. We establish existence, uniqueness, measurability and integrability results for these random entropy solutions, generalizing [28,29] to possibly degenerate hyperbolic-parabolic problems with random data. We next address the numerical approximation of random entropy solutions, specifically the approximation of the deterministic first and second order statistics. To this end, we consider explicit and implicit time discretization and Finite Difference methods in space, and single as well as Multi-Level Monte-Carlo methods to sample the statistics. We establish convergence rate estimates with respect to the discretization parameters, as well as with respect to the overall work, indicating substantial gains in efficiency are afforded under realistic regularity assumptions by the use of the Multi-Level Monte-Carlo method. Numerical experiments are presented which confirm the theoretical convergence estimates.

Multilevel Monte Carlo finite volume methods for random conservation laws with discontinuous flux

“…We first introduce some preliminary concepts which are needed in the exposition. To that end, we follow [34] and [53], see also [32,Sec. 2] and [13,Sec.…”

“…such a framework was developed in a series of papers allowing for random initial datum [40], random (spatially independent) flux [39], and even random source terms [41] and random diffusion [32].…”

Correction to: Multilevel Monte Carlo front-tracking for random scalar conservation laws

“…To render the presentation selfcontained, we recapitulate the relevant definitions as used, e.g., in [2]. For a general Banach space E, the type of the Banach space is defined as follows (see, e.g., [3,Page 246]).…”

Analytical and approximate solution of two-dimensional convection-diffusion problems

“…And some numerical solutions have been developed to solve these types of convection-diffusion problems. likes: Higher-Order ADI method [10] or rational high-order compact ADI method [11], the alternating direction implicit method [12], the finite element method [13], fourth-order compact finite difference method [14], decomposition Method [15], the finite difference method [16], restrictive taylors approximation [17], The fundamental solution [18], finite difference method [19], combined compact difference scheme and alternating direction implicit method [20], higher order compact schemes method [21], the finite volume method [22], the finite difference and legendre spectral method [23] and even the Monte *Corresponding Author Carlo method [24]. Keskin in [25] proposed the RDTM to solve various PDE and fractional nonlinear partial differential equations.…”