A finite volume method for scalar conservation laws with stochastic time-space dependent flux function

We propose a new finite volume method for scalar conservation laws with stochastic time–space dependent flux functions. The stochastic effects appear in the flux function and can be interpreted as a random manner to localize the discontinuity in the time–space dependent flux function. The location of the interface between the fluxes can be obtained by solving a system of stochastic differential equations for the velocity fluctuation and displacement variable. In this paper we develop a modified Rusanov method for the reconstruction of numerical fluxes in the finite volume discretization. To solve the system of stochastic differential equations for the interface we apply a second-order Runge–Kutta scheme. Numerical results are presented for stochastic problems in traffic flow and two-phase flow applications. It is found that the proposed finite volume method offers a robust and accurate approach for solving scalar conservation laws with stochastic time–space dependent flux functions

Hyperbolic Balance Laws: modeling, analysis, and numerics (hybrid meeting)

This workshop brought together leading experts, as well as the most promising young researchers, working on nonlinear hyperbolic balance laws. The meeting focused on addressing new cutting-edge research in modeling, analysis, and numerics. Particular topics included ill-/well-posedness, randomness and multiscale modeling, flows in a moving domain, free boundary problems, games and control

Numerical Methods for Hyperbolic Partial Differential Equations

Department of Mathematical SciencesIn this dissertation, new numerical methods are proposed for different types of hyperbolic partial differential equations (PDEs). The objectives of these developments aim for the improvements in accuracy, robustness, efficiency, and reduction of the computational cost. The dissertation consists of two parts. The first half discusses shock-capturing methods for nonlinear hyperbolic conservation laws, and proposes a new adaptive weighted essentially non-oscillatory WENO-?? scheme in the context of finite difference. Depending on the smoothness of the large stencils used in the reconstruction of the numerical flux, a parameter ?? is set adaptively to switch the scheme between a 5th-order upwind or 6th-order central discretization. A new indicator depending on parameter ?? measures the smoothness of the large stencils in order to choose a smoother one for the reconstruction procedure. ?? is devised based on the possible highest-order variations of the reconstructing polynomials in an L2 sense. In addition, a new set of smoothness indicators ??_k???s of the sub-stencils is introduced. These are constructed in a central sense with respect to the Taylor expansions around point x_j . Numerical results show that the new scheme combines good properties of both 5th-order upwind and 6th-order central schemes. In particular, the new scheme captures discontinuities and resolves small-scaled structures much better than other 5th-order schemes; overcomes the loss of resolution near some critical regions; and is able to maintain symmetry which are drawbacks detected in other 6th-order central WENO schemes. The second part extends the scope to hyperbolic PDEs with uncertainty, and semi-analytical methods using singular perturbation analysis for dispersive PDEs. For the former, a hybrid operator splitting method is developed for computation of the two-dimensional transverse magnetic Maxwell equations in media with multiple random interfaces. By projecting the solutions into random space using the Polynomial Chaos (PC) expansions, the deterministic and random parts of the solution are solved separately. The deterministic parts are then numerically approximated by the FDTD method with domain decomposition implemented on a staggered grid. Statistic quantities are obtained by the Monte Carlo sampling in the post-processing stage. Parallel computing is proposed for which the computational cost grows linearly with the number of random interfaces. The last section deals with spectral methods for dispersive PDEs. The Kortewegde Vries (KdV) equation is chosen as a prototype. By Fourier series, the PDE is transformed into a system of ODEs which is stiff, that is, there are rapid oscillatory modes for large wavenumbers. A new semi-analytical method is proposed to tackle the difficulty. The new method is based on the classical integrating factor (IF) and exponential time differencing (ETD) schemes. The idea is to approximate analytically the stiff parts by the so-called correctors and numerically the non-stiff parts by the IF and ETD methods. It turns out that rapid oscillations are well absorbed by our corrector method, yielding better accuracy in the numerical results. Due to the nonlinearity, all Fourier modes interact with each other, causing the computation of the correctors to be very costly. In order to overcome this, the correctors are recursively constructed to accurately capture the stiffness of the mode interactions.ope

ADAPTIVE GRID BASED FINITE DIFFERENCE METHODS FOR SOLUTION OF HYPERBOLIC PDES: APPLICATION TO COMPUTATIONAL MECHANICS AND UNCERTAINTY QUANTIFICATION

Novel finite-difference based numerical methods for solution of linear and nonlinear hyperbolic partial differential equations (PDEs) using adaptive grids are proposed in this dissertation. The overall goal of this research is to improve the accuracy and/or computational efficiency of numerical solutions via the use of adaptive grids and suitable modifications of a given low-order order finite-difference scheme. These methods can be grouped in two broad categories. The first category of adaptive FD methods proposed in the dissertation attempt to reduce the truncation error and/or enhance the accuracy of the underlying numerical schemes via grid distribution alone. Some approaches for grid distribution considered include those based on (i) a moving uniform mesh/domain, (ii) adaptive gradient based refinement (AGBR) and (iii) unit local Courant-Freidrich-Lewy (CFL) number. The improvement in the accuracy which is obtained using these adaptive methods is limited by the underlying scheme formal order of accuracy. In the second category, the CFL based approach proposed in the first category was extended further using defect correction in order to improve the formal order of accuracy and computational efficiency significantly (i.e. by at least one order or higher). The proposed methods in this category are constructed based upon the analysis of the leading order error terms in the modified differential equation associated with the underlying partial differential equation and finite difference scheme. The error terms corresponding to regular and irregular perturbations are identified and the leading order error terms associated with regular perturbations are eliminated using a non-iterative defect correction approach while the error terms associated with irregular perturbations are eliminated using grid adaptation. In the second category of methods involving defect correction (or reduction of leading order terms of truncation error), we explored two different approaches for selection of adaptive grids. These are based on (i) optimal grid dis- tribution and (ii) remapping with monotonicity preserving interpolation. While the first category of methods may be preferred in view of ease of implementation and lower computational complexity, the second category of methods may be preferred in view of greater accuracy and computational efficiency. The two broad categories of methods, which have been applied to problems involving both bounded and unbounded domains, were also extended to multidimensional cases using a dimensional splitting approaches. The performance of these methods was demonstrated using several example problems in computational uncertainty quantification (CUQ) and computational mechanics. The results of the application of the proposed approaches all indicate improvement in both the accuracy and computational efficiency (by about three orders of magnitude in some selected cases) of underlying schemes. In the context of CUQ, all three proposed adaptive finite different solvers are combined with the Gauss-quadrature sampling technique in excitation space to obtain statistical quantities of interest for dynamical systems with parametric uncertainties from the solution of Liouville equation, which is a linear hyperbolic PDE. The numerical results for four canonical UQ problems show both enhanced computational efficiency and improved accuracy of the proposed adaptive FD solution of the Liouville equation compared to its standard/fixed domain FD solutions. Moreover, the results for canonical test problems in computational mechanics indicate that the proposed approach for increasing the formal order of the underlying FD scheme can be easily implemented in multidimensional spaces and gives an oscillation-free numerical solution with a desired order of accuracy in a reasonable computational time. This approach is shown to provide a better computational time compared to both the underlying scheme (by about three orders of magnitude) and standard FD methods of the same order of accuracy

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal