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We introduce the spatial discretization schemes for systems of conservation laws that we use later. For smooth problems, summation-by-parts (SBP) operators with weak enforcement of boundary conditions (SAT) are presented. The SBP-SAT schemes allow for the design of stable high-order accurate schemes. Summation by parts is the discrete equivalent of integration by parts and the matrix operators that are presented lead to energy estimates that in turn lead to provable stability. The semidiscrete stability follows naturally from the continuous analysis of wellposedness which provides the boundary conditions in the SBP-SAT technique.Stability and boundary conditions are the main reason for choosing to use SBP operators. Provable stability means that numerical convergence to the true solution can be guaranteed. There are many alternative numerical schemes that appear to converge, but for the stochastic Galerkin formulations of interest here, we want to be able to prove stability in situations that would otherwise be hard to handle. An example is a solution with multiple discontinuities crossing the numerical boundary. That situation requires stability and correct imposition of boundary conditions. For non-smooth problems, the need to accurately capture multiple solution discontinuities of hyperbolic stochastic Galerkin systems calls for shock-capturing methods. We outline how the use of the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) with flux limiters and the HLL (after Harten, Lax and van Leer) Riemann solver can be used to treat these cases. We also discuss in brief how to add artificial dissipation and an issue regarding time-integration.The problems presented here can all be written as one-dimensional conservation laws,

In this chapter we review methods for formulating partial differential equations based on the random field representations outlined in Chap. 2. These include the stochastic Galerkin method, which is the predominant choice in this book, as well as other methods that frequently occur in the literature. We also briefly discuss methods that are not polynomial chaos methods themselves but are viable alternatives. Intrusive MethodsIn the context of gPC, problem formulations result in a new set of equations that are distinctly different from the original set of equations and thus require the design of new numerical solvers. These solvers are referred to as intrusive methodsas opposed to non-intrusive stochastic methods that exclusively rely on existing deterministic codes. Stochastic Galerkin MethodsThe stochastic Galerkin method was introduced by Ghanem and Spanos in order to solve linear stochastic equations [11]. It relies on a weak problem formulation where the set of solution basis functions (trial functions) is the same as the space of stochastic test functions. Consider a general scalar conservation law defined on a spatial domain˝x with boundary x subject to initial and boundary conditions, given by @u.x; t; / @t C @f .u.

The aim of this chapter, based on [18], is to present accurate and stable numerical schemes for the solution of a class of linear diffusive transport problems. The advection-diffusion equation subject to uncertain viscosity with known statistical description is represented by a spectral expansion in the stochastic dimension. The gPC framework and the stochastic Galerkin method are used to obtain an extended system which is analyzed to find discretization constraints on monotonicity, stiffness and stability. A comparison of stochastic Galerkin versus methods based on repeated evaluations of deterministic solutions, such as stochastic collocation, is provided but this is not our primary focus. However, we do include a few examples on relative performance and numerical properties with respect to monotonicity requirements and convergence to steady-state, to encourage the use of stochastic Galerkin methods.Special care is exercised to ensure that the stochastic Galerkin projection results in a system with a positive semidefinite diffusion matrix. The sign of the eigenvalues of a pure advection problem is not a stumbling block as long as the boundary conditions are properly adjusted to match the number of ingoing characteristics, as shown in [7]. Unlike the case of stochastic advection, the sign of the eigenvalues of the diffusion matrix of the advection-diffusion problem is crucial. A negative eigenvalue leads to the growth of the solution norm and hence numerical instability. The source of the growth is in the volume term, and no treatment of the boundary conditions can eliminate it.Advection-diffusion problems with uncertainty have been investigated by several authors. Ghanem and Dham

Burgers' equation is a non-linear model problem from which many results can be extended to other hyperbolic systems, e.g., the Euler equations. In this chapter, a detailed uncertainty quantification analysis is performed for the Burgers' equation; we employ a spectral representation of the solution in the form of polynomial chaos expansion. The PDE is stochastic as a result of the uncertainty in the initial and boundary values. Stochastic Galerkin projection results in a coupled, deterministic system of nonlinear hyperbolic equations from which statistics of the solution can be determined.Previous investigations on the effect of uncertainty on Burgers' equation focused on the location of the transition layer of a shock discontinuity arising in simulations of the Burgers' equation with nonzero viscosity. Small, one-sided perturbations imply large variation in the location of the transition layer, so-called supersensitivity [15], which has been shown to be a problem in deterministic as well as stochastic simulations. The results from the polynomial chaos approach were accurate and the method was faster than the Monte Carlo method [14,15]. Burgers' equation with a stochastic forcing term has also been investigated and compared to standard Monte Carlo methods [6].In this chapter, based on [12], we perform a fundamental analysis of the Burgers' equation and develop a numerical framework to study the effect of uncertainty in the boundary conditions. With the assumption that the uncertainty of the boundary data has a Gaussian distribution we allow the occurrence of unbounded solutions. Assuming that the boundary data resemble the Gaussian distribution but are bounded to a sufficiently large range does not alter the numerical results. Convergence is proven by a suitable choice of functional space.In order to ensure stability of the discretized system of equations, SBP operators and weak imposition of boundary conditions [2,10,11] are used to obtain energy estimates, as demonstrated in Chap. 5. The system is expressed in a split form that combines the conservative and non-conservative formulation [9]. A particular set of