Recent gain of interest in discontinuous Galerkin (DG) methods shows their success in computational fluid dynamics. One potential drawback is the high number of globally coupled unknowns. By means of hybridization, th...
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ISBN:
(纸本)9783319198002;9783319197999
Recent gain of interest in discontinuous Galerkin (DG) methods shows their success in computational fluid dynamics. One potential drawback is the high number of globally coupled unknowns. By means of hybridization, this number can be significantly reduced. The hybridized DG (HDG) method has proven to be beneficial especially for steady flows. In this work we apply it to a time-dependent flow problem with shocks. Due to its inherently implicit structure, time integration methods such as diagonally implicit Runge-Kutta (DIRK) methods present themselves as natural candidates. Furthermore, as the application of flux limiting to HDG is not straightforward, an artificial viscosity model is applied to stabilize the method.
For functions u is an element of H-1 (Omega) in an open, bounded polyhedron Omega subset of R-d of dimension d = 1, 2, 3, which are analytic in (Omega) over bar /S with point singularities concentrated at the set S su...
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ISBN:
(纸本)9783319198002;9783319197999
For functions u is an element of H-1 (Omega) in an open, bounded polyhedron Omega subset of R-d of dimension d = 1, 2, 3, which are analytic in (Omega) over bar /S with point singularities concentrated at the set S subset of (Omega) over bar consisting of a finite number of points in (Omega) over bar, the exponential rate exp(-b(d+1)root N) of convergence of hp-version continuous Galerkin finite element methods on families of regular, simplicial meshes in Omega can be achieved. The simplicial meshes are assumed to be geometrically refined towards S and to be shape regular, but are otherwise unstructured.
Typically when a semi-discrete approximation to a partial differential equation (PDE) is constructed a discretization of the spatial operator with a truncation error tau is derived. This discrete operator should be se...
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ISBN:
(纸本)9783319198002;9783319197999
Typically when a semi-discrete approximation to a partial differential equation (PDE) is constructed a discretization of the spatial operator with a truncation error tau is derived. This discrete operator should be semi-bounded for the scheme to be stable. Under these conditions the Lax-Richtmyer equivalence theorem assures that the scheme converges and that the error will be, at most, of the order of parallel to tau parallel to. In most cases the error is in indeed of the order of parallel to tau parallel to. We demonstrate that for the Heat equation stable schemes can be constructed, whose truncation errors are tau, however, the actual errors are much smaller. This gives more degrees of freedom in the design of schemes which can make them more efficient (more accurate or compact) than standard schemes. In some cases the accuracy of the schemes can be further enhanced using post-processing procedures.
A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations is considered. We use the energy method to derive well-posed boundary conditions for the continuous problem. Summatio...
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ISBN:
(纸本)9783319198002;9783319197999
A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations is considered. We use the energy method to derive well-posed boundary conditions for the continuous problem. Summation-by-Parts (SBP) operators together with a weak imposition of the boundary and initial conditions using Simultaneously Approximation Terms (SATs) guarantee energy-stability of the fully discrete scheme. We construct a time-dependent SAT formulation that automatically imposes the boundary conditions, and show that the numerical Geometric Conservation Law (GCL) holds. Numerical calculations corroborate the stability and accuracy of the approximations. As an application we study the sound propagation in a deforming domain using the linearized Euler equations.
Atmospheric flows are characterized by a large range of length scales as well as strong gradients. The accurate simulation of such flows requires numerical algorithms with high spectral resolution, as well as the abil...
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ISBN:
(纸本)9783319198002;9783319197999
Atmospheric flows are characterized by a large range of length scales as well as strong gradients. The accurate simulation of such flows requires numerical algorithms with high spectral resolution, as well as the ability to provide nonoscillatory solutions across regions of high gradients. These flows exhibit a large range of time scales as well-the slowest waves propagate at the flow velocity and the fastest waves propagate at the speed of sound. Time integration with explicit methods are thus inefficient, although algorithms with semi-implicit time integration have been used successfully in past studies. We propose a finite-difference method for atmospheric flows that uses a weighted compact scheme for spatial discretization and implicit-explicit additive Runge-Kutta methods for time integration. We present results for a benchmark atmospheric flow problem and compare our results with existing ones in the literature.
A robust interface treatment for the discontinuous coefficient advection equation satisfying time-independent jump conditions is presented. The aim of the investigation is to show how the different concepts like well-...
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ISBN:
(纸本)9783319198002;9783319197999
A robust interface treatment for the discontinuous coefficient advection equation satisfying time-independent jump conditions is presented. The aim of the investigation is to show how the different concepts like well-posedness, conservation and stability are related. The equations are discretized using high order finite difference methods on Summation By Parts (SBP) form. The interface conditions are weakly imposed using the Simultaneous Approximation Term (SAT) procedure. Spectral analysis and numerical simulations corroborate the theoretical findings.
Shape optimization is concerned about finding optimal designs under the aspect of some cost criteria often involving the solution of a partial differential equation (PDE) over the afore said unknown shape. In general,...
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ISBN:
(数字)9783319233154
ISBN:
(纸本)9783319233154;9783319233147
Shape optimization is concerned about finding optimal designs under the aspect of some cost criteria often involving the solution of a partial differential equation (PDE) over the afore said unknown shape. In general, industrial cases involve a geometric model from Computer Aided Design (CAD). However, solving PDEs requires an analysis suitable working model, typically a Finite Element (FEM) triangulation. Hence, some of the geometric properties known from the CAD model may be lost during this format change. Therefore, we employ isogeometric analysis (IGA) instead, which has a tighter connection between geometry, simulation and shape optimization. In this paper, we present a self-contained treatment of gradient based shape optimization method with isogeometric analysis, focusing on algorithmic and practical aspects like computation of shape gradients in an IGA formulation and updating B-spline and NURBS geometries.
The present work proposes the extension of the IMERSPEC methodology for numerical simulations of two-phase flows. This methodology consists of the fusion between the Fourier pseudospectral method and the immersed boun...
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ISBN:
(纸本)9783319198002;9783319197999
The present work proposes the extension of the IMERSPEC methodology for numerical simulations of two-phase flows. This methodology consists of the fusion between the Fourier pseudospectral method and the immersed boundary method for non-periodical problems. This method was originally developed for single-phase and incompressible flows (Mariano et al., Comput Model Eng Sci 59: 181-216, 2010). In the present paper, we extend this methodology for two-phase flows using the front-tracking method to model the fluid-fluid interface. The results involving the spurious currents, mass conservation and analysis through numerical experimental bubbles rise, show that the proposed method can be considered validated and promising to computational fluid dynamics (CFD).
We design adaptive high-order Galerkin methods for the solution of linear elliptic problems and study their performance. We first consider adaptive Fourier-Galerkin methods and Legendre-Galerkin methods, which offer u...
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ISBN:
(纸本)9783319198002;9783319197999
We design adaptive high-order Galerkin methods for the solution of linear elliptic problems and study their performance. We first consider adaptive Fourier-Galerkin methods and Legendre-Galerkin methods, which offer unlimited approximation power only restricted by solution and data regularity. Their analysis of convergence and optimality properties reveals a sparsity degradation for Gevrey classes. We next turn our attention to the hp-version of the finite element method, design an adaptive scheme which hinges on a recent algorithm by P. Binev for adaptive hp-approximation, and discuss its optimality properties.
In general, solutions of nonlinear hyperbolic PDEs contain shocks or develop discontinuities. One option for improving the numerical treatment of the spurious oscillations that occur near these artifacts is through th...
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ISBN:
(纸本)9783319198002;9783319197999
In general, solutions of nonlinear hyperbolic PDEs contain shocks or develop discontinuities. One option for improving the numerical treatment of the spurious oscillations that occur near these artifacts is through the application of a limiter. The cells where such treatment is necessary are referred to as troubled cells. In this article, we discuss the multiwavelet troubled-cell indicator that was introduced by Vuik and Ryan (J Comput Phys 270: 138-160, 2014). We focus on the relation between the highest-level multiwavelet coefficients and jumps in (derivatives of) the DG approximation. Based on this information, we slightly modify the originalmultiwavelet troubled-cell indicator. Furthermore, we show one-dimensional test cases using the modified multiwavelet troubled-cell indicator.
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