Mohammed Seaïd

Well-balanced finite volume schemes for pollutant transport by shallow water equations on unstructured meshes

Pollutant transport by shallow water flows on non-flat topography is presented and numerically solved using a finite volume scheme. The method uses unstructured meshes, incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep bathymetric gradients that may form in the approximate solution. The scheme is non-oscillatory and possesses conservation property that conserves the pollutant mass during the transport process. Numerical results are presented for three test examples which demonstrate the accuracy and robustness of the scheme and its applicability in predicting pollutant transport by shallow water flows. In this paper, we also apply the developed scheme for a pollutant transport event in the Strait of Gibraltar. The scheme is efficient, robust and may be used for practical pollutant transport phenomena.

NUMERICAL METHODS AND OPTIMAL CONTROL FOR GLASS COOLING PROCESSES

ABSTRACT In this paper, we discuss numerical and analytical approximations of radiative heat transfer equations used to model cooling processes of molten glass. Simplified diffusion type approximations are discussed and investigated numerically. These approximations are also used to develop acceleration methods for the iterative solution of the full radiative heat transfer problem. Moreover, applications of the above diffusion type approximations to optimal control problems for glass cooling processes are discussed.

Higher-order relaxation schemes for hyperbolic systems of conservation laws

We present a higher order generalization for relaxation methods in the framework presented by Jin and Xin in [10]. The schemes employ general higher order integration for spatial discretization and higher order implicit-explicit (IMEX) schemes or Total Variation diminishing (TVD) Runge–Kutta schemes for time integration of relaxing or relaxed schemes, respectively, for time integration. Numerical experiments are performed on various test problems, in particular, the Burgers and Euler equations of inviscid gas dynamics in both one and two space dimensions. In addition, uniform convergence with respect to the relaxation parameter is demonstrated.

Radiation models for thermal flows at low Mach number

Simplified approximate models for radiation are proposed to study thermal effects in low Mach flow in open tunnels. The governing equations for fluid dynamics are derived by applying a low Mach asymptotic in the compressible Navier-Stokes problem. Based on an asymptotic analysis we show that the integro-differential equation for radiative transfer can be replaced by a set of differential equations which are independent of angle variable and easy to solve using standard numerical discretizations. As an application we consider a simplified fire model in vehicular tunnels. The results presented in this paper show that the proposed models are able to predict temperature in the tunnels accurately with low computational cost.

A comparison of approximate models for radiation in gas turbines

Approximate equations for radiative heat transfer equations coupled to an equation for the temperature are stated and a comparative numerical study of the different approximations is given. The approximation methods considered here range from moment methods to simplified PN-approximations. Numerical experiments and comparisons in different space dimensions and for various physical situations are presented.

Solution of the Sediment Transport Equations Using a Finite Volume Method Based on Sign Matrix

We present a finite volume method for the numerical solution of the sediment transport equations in one and two space dimensions. The numerical fluxes are reconstructed using a modified Roe scheme that incorporates, in its reconstruction, the sign of the Jacobian matrix in the sediment transport system. A well-balanced discretization is used for the treatment of source terms. The method is well balanced, nonoscillatory, and suitable for both structured and unstructured triangular meshes. An adaptive procedure is also considered for the two-dimensional problems to update the bed-load accounting for the interaction between the bed-load and the water flow. The proposed method is applied to several sediment transport problems in one and two space dimensions. The numerical results demonstrate high resolution of the proposed finite volume method and confirm its capability to provide accurate simulations for sediment transport problems under flow regimes with strong shocks.

A simple finite volume method for the shallow water equations

We present a new finite volume method for the numerical solution of shallow water equations for either flat or non-flat topography. The method is simple, accurate and avoids the solution of Riemann problems during the time integration process. The proposed approach consists of a predictor stage and a corrector stage. The predictor stage uses the method of characteristics to reconstruct the numerical fluxes, whereas the corrector stage recovers the conservation equations. The proposed finite volume method is well balanced, conservative, non-oscillatory and suitable for shallow water equations for which Riemann problems are difficult to solve. The proposed finite volume method is verified against several benchmark tests and shows good agreement with analytical solutions.

Method of lines for stochastic boundary-value problems with additive noise

Abstract We propose a new numerical method for solving stochastic boundary-value problems. The method uses the deterministic method of lines to treat the time, space and randomness separately. The emphasis in the present study is given to stochastic partial differential equations with forced additive noise. The spatial discretization is carried out using a second-order finite volume method, while the associated stochastic differential system is numerically solved using a class of stochastic Runge–Kutta methods. The performance of the proposed methods is tested for a stochastic advection–diffusion problem and a stochastic Burgers equation driven with white noise. Numerical results are presented in both one and two space dimensions.

Efficient Preconditioning of Linear Systems Arising from the Discretization of Radiative Transfer Equation

Preconditioning techniques for iterative solvers of discretized radiative transfer equation are presented. Discrete ordinates collocation and Diamond differencing are used for angle and space discretizations respectively. To solve the resulting linear system we formulate source iteration, diffusion synthetic acceleration and Krylov subspace methods. We also introduce a fast multilevel algorithm. All these methods can be viewed as preconditioned iterative methods with different preconditioners. Numerical results along with comparisons of effectiveness and efficiency of these solvers are carried out on several test problems with both continuous and discontinuous variables.

Semi-lagrangian integration schemes for viscous incompressible flows

Abstract A new second-order accurate scheme for the computation of unsteady viscous incompressible flows is proposed. The scheme is based on the vorticity-stream function formulation along the characteristics and consists of combining the modified method of characteristics with an explicit scheme with an extended real stability interval. A comparison of the new method with the semi-Lagrangian Cranck-Nicolson and classical semi-Lagrangian Runge-Kutta schemes is presented. Numerical results are carried out on Navier-Stokes equations and this efficient second-order scheme has also made it possible to compute the driven cavity ow at a high Reynolds number on a refined grid at a reasonable cost. The procedure can be generalized to more than two dimensions.