Numerical analysis of the Burgers’ equation in the presence of uncertainty

Journal of Computational Physics

Wahlsten

2015

Self Cite

We consider a hyperbolic system in one space dimension with uncertainty in the boundary and initial data. Our aim is to show that different boundary conditions gives different convergence rates of the variance of the solution. This means that we can with the same knowledge of data get a more or less accurate description of the uncertainty in the solution. A variety of boundary conditions are compared and both analytical and numerical estimates of the variance of the solution is presented. As an application, we study the effect of this technique on a subsonic outflow boundary for the Euler equations

Section: Introductionmentioning

confidence: 99%

Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations

Journal of Computational Physics

Wahlsten

2015

Self Cite

“…The possibility of applying the SBP-SAT technique to the coupled PDEs resulting from the use of polynomial chaos in combination with a stochastic Galerkin projection is shown in Pettersson et al [19,20]. …”

Section: Remark 32mentioning

confidence: 99%

The effect of uncertain geometries on advection–diffusion of scalar quantities

Wahlsten

2017

Bit Numer Math

Self Cite

The two dimensional advection-diffusion equation in a stochastically varying geometry is considered. The varying domain is transformed into a fixed one and the numerical solution is computed using a high-order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. Statistics of the solution are computed non-intrusively using quadrature rules given by the probability density function of the random variable. As a quality control, we prove that the continuous problem is strongly well-posed, that the semi-discrete problem is strongly stable and verify the accuracy of the scheme. The technique is applied to a heat transfer problem in incompressible flow. Statistical properties such as confidence intervals and variance of the solution in terms of two functionals are computed and discussed. We show that there is a decreasing sensitivity to geometric uncertainty as we gradually lower the frequency and amplitude of the randomness. The results are less sensitive to variations in the correlation length of the geometry.

“…In a previous paper [12], we studied the dependency of the solution on the order of truncation; here we will consider expansions of order M = 1. Expectation and variance can be expressed in terms of the polynomial chaos coefficients, as…”

Section: Problem Setupmentioning

confidence: 99%

“…The application of the polynomial chaos approach leads to a new hyperbolic systems with multiple discontinuities [12]. The weak resemblance to the corresponding deterministic problem suggests an appropriate way to specify boundary conditions for the solution mean, but gives no concrete information on the treatment of higher order moments.…”

Section: Introductionmentioning

confidence: 99%

Boundary procedures for the time-dependent Burgers' equation under uncertainty

Pettersson

Acta Mathematica Scientia

Iaccarino

2010

Self Cite

The Burgers' equation with uncertain initial and boundary conditions is approximated using a Polynomial Chaos Expansion (PCE) approach where the solution is represented as a series of stochastic, orthogonal polynomials. The resulting truncated PCE system is solved using a novel numerical discretization method based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. The resulting PCE solution yields an accurate quantitative description of the stochastic evolution of the system, provided that appropriate boundary conditions are available. The specification of the boundary data is shown to influence the solution; we will discuss the problematic implications of the lack of precisely characterized boundary data and possible ways of imposing stable and accurate boundary conditions.