A Dirichlet problem is considered for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter epsilon (epsilon epsilon (0, 1]) multiplying the highest-order derivative ...
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ISBN:
(纸本)9783319257273;9783319257259
A Dirichlet problem is considered for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter epsilon (epsilon epsilon (0, 1]) multiplying the highest-order derivative in the equation. This problem is approximated by the standard monotone finite difference scheme on a uniform grid. Such a scheme does not converge epsilon-uniformly in the maximum norm when the number of grid nodes grows. Moreover, under its convergence, the scheme is not epsilon-uniformly well conditioned and stable to data perturbations of the discrete problem and/or computer perturbations. For small values of epsilon, perturbations of the grid solution can significantly exceed (and even in order of magnitude) the error in the unperturbed solution. For a computer difference scheme (the standard scheme in the presence of computer perturbations), technique is developed for theoretical and experimental study of convergence of perturbed grid solutions. For computer perturbations, conditions are obtained (depending on the parameter epsilon and the number of grid intervals N), for which the solution of the computer scheme converges in the maximum norm with the same order as the solution of the standard scheme in the absence of perturbations.
This paper describes extensions of the generalized summation-by-parts (GSBP) framework to the approximation of the second derivative with a variable coefficient and to time integration. GSBP operators for the second d...
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ISBN:
(纸本)9783319198002;9783319197999
This paper describes extensions of the generalized summation-by-parts (GSBP) framework to the approximation of the second derivative with a variable coefficient and to time integration. GSBP operators for the second derivative lead to more efficient discretizations, relative to the classical finite-difference SBP approach, as they can require fewer nodes for a given order of accuracy. Similarly, for time integration, time-marching methods based on GSBP operators can be more efficient than those based on classical SBP operators, as they minimize the number of solution points which must be solved simultaneously. Furthermore, we demonstrate the link between GSBP operators and Runge-Kutta time-marching methods.
We develop stable finite difference approximations for a multi-physics problem that couples elastic wave propagation in one domain to acoustic wave propagation in another domain. The approximation consists of one fini...
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ISBN:
(纸本)9783319198002;9783319197999
We develop stable finite difference approximations for a multi-physics problem that couples elastic wave propagation in one domain to acoustic wave propagation in another domain. The approximation consists of one finite difference scheme in each domain together with discrete interface conditions that couple the two schemes. The finite difference approximations use summation-by-parts (SBP) operators, which lead to stability of the coupled problem. Furthermore, we develop a new way to enforce boundary conditions for SBP discretizations of first order problems. The new method, which uses ghost points to enforce the boundary conditions, is a flexible alternative to the more established projection and SAT methods.
A characterisation theorem for best uniform wavenumber approximations by central difference schemes is presented. A central difference stencil is derived based on the theorem and is compared with dispersion relation p...
ISBN:
(纸本)9783319198002;9783319197999
A characterisation theorem for best uniform wavenumber approximations by central difference schemes is presented. A central difference stencil is derived based on the theorem and is compared with dispersion relation preserving schemes and with classical central differences for a relevant test problem.
We consider the SUPG method for the numerical solution of the scalar steady convection-diffusion equation using conforming simplicial piecewise linear finite elements. We change the convective vector in the SUPG stabi...
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ISBN:
(纸本)9783319257273;9783319257259
We consider the SUPG method for the numerical solution of the scalar steady convection-diffusion equation using conforming simplicial piecewise linear finite elements. We change the convective vector in the SUPG stabilizing term and adjust the triangulation so that the discrete maximum principle is satisfied. Then the error analysis is performed and the method is tested on several numerical examples.
We present and analyze a Lagrange-Galerkin (LG) method combined with a local projection stabilization (LPS) technique for convection dominated convection-diffusion-reaction equations. This type of stabilization improv...
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ISBN:
(纸本)9783319257273;9783319257259
We present and analyze a Lagrange-Galerkin (LG) method combined with a local projection stabilization (LPS) technique for convection dominated convection-diffusion-reaction equations. This type of stabilization improves the accuracy and performance of conventional LG methods when the diffusion coefficient is very small. Numerical tests support the results of the numerical error analysis.
This chapter presents a constructive derivation of HDG methods for convection-diffusion-reaction equation using the Rankine-Hugoniot condition. This is possible due to the fact that, in the first order form, convectio...
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ISBN:
(纸本)9783319198002;9783319197999
This chapter presents a constructive derivation of HDG methods for convection-diffusion-reaction equation using the Rankine-Hugoniot condition. This is possible due to the fact that, in the first order form, convection-diffusion-reaction equation is a hyperbolic system. As such it can be discretized using the standard upwind DG method. The key is to realize that the Rankine-Hugoniot condition naturally provides an upwind HDG framework. The chief idea is to first break the uniqueness of the upwind flux across element boundaries by introducing single-valued new trace unknowns on the mesh skeleton, and then re-enforce the uniqueness via algebraic conservation constraints. Essentially, the HDG framework is a redesign of the standard DG approach to reduce the number of coupled unknowns. In this work, an upwind HDG method with one trace unknown is systematically constructed, and then extended to a family of penalty HDG schemes. Various existing HDG methods are rediscovered using the proposed framework.
We develop a high performance computing (HPC) framework for efficient simulations of a class of fractional-order partial differential equations (FPDE), using high-order in time and space parallel algorithms. HPC syste...
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ISBN:
(纸本)9783319198002;9783319197999
We develop a high performance computing (HPC) framework for efficient simulations of a class of fractional-order partial differential equations (FPDE), using high-order in time and space parallel algorithms. HPC systems provide a large number of processing cores with limitations on the amount of memory available per core. Such limitations impose severe constraints for resolving fine spatial structures that require large degrees of freedom (DoF). In this article, using several message passing interface (MPI) communicators, we develop and demonstrate an efficient hybrid framework that combines parallel in time and space tasks that facilitate careful balance between parallel performance within the memory constraint to simulate the FPDE model. We demonstrate the approach for a 3D fractional PDE using several million spatial DoF.
In the context of stabilization of high order spectral elements, we introduce a dissipative scheme based on the solution of the compressible Euler equations that are regularized through the addition of a residual-base...
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ISBN:
(纸本)9783319198002;9783319197999
In the context of stabilization of high order spectral elements, we introduce a dissipative scheme based on the solution of the compressible Euler equations that are regularized through the addition of a residual-based stress tensor. Because this stress tensor is proportional to the residual of the unperturbed equations, its effect is close to none where the solution is sufficiently smooth, whereas it increases elsewhere. This paper represents a first extension of the work by Nazarov and Hoffman (Int J Numer Methods Fluids 71:339-357, 2013) to high-order spectral elements in the context of low Mach number atmospheric dynamics. The simulations show that the method is reliable and robust for problems with important stratification and thermal processes such as the case of moist convection. The results are partially compared against a Smagorinsky solution. With this work we mean to make a step forward in the implementation of a stabilized, high order, spectral element large eddy simulation (LES) model within the Nonhydrostatic Unified Model of the Atmosphere, NUMA.
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