Search results for "Computational Mathematic"
showing 10 items of 987 documents
Capturing Shock Reflections: An Improved Flux Formula
1996
Godunov type schemes, based on exact or approximate solutions to the Riemann problem, have proven to be an excellent tool to compute approximate solutions to hyperbolic systems of conservation laws. However, there are many instances in which a particular scheme produces inappropriate results. In this paper we consider several situations in which Roe's scheme gives incorrect results (or blows up all together) and we propose an alternative flux formula that produces numerical approximations in which the pathological behavior is either eliminated or reduced to computationally acceptable levels.
Power ENO methods: a fifth-order accurate Weighted Power ENO method
2004
In this paper we introduce a new class of ENO reconstruction procedures, the Power ENO methods, to design high-order accurate shock capturing methods for hyperbolic conservation laws, based on an extended class of limiters, improving the behavior near discontinuities with respect to the classical ENO methods. Power ENO methods are defined as a correction of classical ENO methods [J. Comput. Phys. 71 (1987) 231], by applying the new limiters on second-order differences or higher. The new class of limiters includes as a particular case the minmod limiter and the harmonic limiter used for the design of the PHM methods [see SIAM J. Sci. Comput. 15 (1994) 892]. The main features of these new ENO…
A Flux-Split Algorithm Applied to Relativistic Flows
1998
The equations of RFD can be written as a hyperbolic system of conservation laws by choosing an appropriate vector of unknowns. We give an explicit formulation of the full spectral decomposition of the Jacobian matrices associated with the fluxes in each spatial direction, which is the essential ingredient of the techniques we propose in this paper. These techniques are based on the recently derived flux formula of Marquina, a new way to compute the numerical flux at a cell interface which leads to a conservative, upwind numerical scheme. Using the spectral decompositions in a fundamental way, we construct high order versions of the basic first-order scheme described by R. Donat and A. Marqu…
High-order Runge–Kutta–Nyström geometric methods with processing
2001
Abstract We present new families of sixth- and eighth-order Runge–Kutta–Nystrom geometric integrators with processing for ordinary differential equations. Both the processor and the kernel are composed of explicitly computable flows associated with non trivial elements belonging to the Lie algebra involved in the problem. Their efficiency is found to be superior to other previously known algorithms of equivalent order, in some case up to four orders of magnitude.
On the numerical solution of axisymmetric domain optimization problems by dual finite element method
1994
Shape optimization of an axisymmetric three-dimensional domain with an elliptic boundary value state problem is solved. Since the cost functional is given in terms of the cogradient of the solution, a dual finite element method based on the minimum of complementary energy principle is used. © 1994 John Wiley & Sons, Inc.
Convergence of a finite volume scheme for the compressible Navier–Stokes system
2019
We study convergence of a finite volume scheme for the compressible (barotropic) Navier–Stokes system. First we prove the energy stability and consistency of the scheme and show that the numerical solutions generate a dissipative measure-valued solution of the system. Then by the dissipative measure-valued-strong uniqueness principle, we conclude the convergence of the numerical solution to the strong solution as long as the latter exists. Numerical experiments for standard benchmark tests support our theoretical results.
The MAST FV/FE scheme for the simulation of two-dimensional thermohaline processes in variable-density saturated porous media
2009
A novel methodology for the simulation of 2D thermohaline double diffusive processes, driven by heterogeneous temperature and concentration fields in variable-density saturated porous media, is presented. The stream function is used to describe the flow field and it is defined in terms of mass flux. The partial differential equations governing system is given by the mass conservation equation of the fluid phase written in terms of the mass-based stream function, as well as by the advection-diffusion transport equations of the contaminant concentration and of the heat. The unknown variables are the stream function, the contaminant concentration and the temperature. The governing equations sy…
Asymptotic preserving IMEX finite volume schemes for low Mach number Euler equations with gravitation
2017
In this paper we will present and analyze a new class of the IMEX finite volume schemes for the Euler equations with a gravity source term. We will in particular concentrate on a singular limit of weakly compressible flows when the Mach number M1. In order to efficiently resolve slow dynamics we split the whole nonlinear system in a stiff linear part governing the acoustic and gravity waves and a non-stiff nonlinear part that models nonlinear advection effects. For time discretization we use a special class of the so-called globally stiffly accurate IMEX schemes and approximate the stiff linear operator implicitly and the non-stiff nonlinear operator explicitly. For spatial discretization t…
A normalized iterative Smoothed Particle Hydrodynamics method
2020
Abstract In this paper we investigate on a normalized iterative approach to improve the Smoothed Particle Hydrodynamics (SPH) estimate of a function. The method iterates on the residuals of an initial SPH approximation to obtain a more accurate solution. The iterative strategy preserves the matrix-free nature of the method, does not require changes on the kernel function and it is not affected by disordered data distribution. The iterative refinement is further improved by ensuring linear approximation order to the starting iterative values. We analyze the accuracy and the convergence of the method with the standard and normalized formulation giving evidence of the enhancements obtained wit…
Inversion Formulas for the Discretized Hilbert Transform on the Unit Circle
1998
A discrete version of the Hilbert transform on the unit circle is considered. Its Moore--Penrose inverse with respect to suitable scalar products is derived for different side conditions. Furthermore, stability of the pseudo-inverse is studied. These results allow the efficient computation of approximate solutions of singular integral equations with Hilbert kernel. Furthermore, the stability analysis of such methods becomes much easier even for graded meshes which are useful for weakly singular solutions.