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1.
Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators 下载免费PDF全文
High order discretization schemes play more important role in fractional operators
than classical ones. This is because usually for classical derivatives the stencil
for high order discretization schemes is wider than low order ones; but for fractional
operators the stencils for high order schemes and low order ones are the same. Then
using high order schemes to solve fractional equations leads to almost the same computational
cost as first order schemes but the accuracy is greatly improved. Using
the fractional linear multistep methods, Lubich obtains the ν-th order (ν≤6) approximations
of the α-th derivative (α>0) or integral (α<0) [Lubich, SIAM J. Math. Anal.,
17, 704-719, 1986], because of the stability issue the obtained scheme can not be directly
applied to the space fractional operator with α∈(1,2) for time dependent problem. By
weighting and shifting Lubich's 2nd order discretization scheme, in [Chen & Deng,
SINUM, arXiv:1304.7425] we derive a series of effective high order discretizations for
space fractional derivative, called WSLD operators there. As the sequel of the previous
work, we further provide new high order schemes for space fractional derivatives
by weighting and shifting Lubich's 3rd and 4th order discretizations. In particular,
we prove that the obtained 4th order approximations are effective for space fractional
derivatives. And the corresponding schemes are used to solve the space fractional
diffusion equation with variable coefficients. 相似文献
2.
Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis 下载免费PDF全文
David Kamensky John A. Evans & Ming-Chen Hsu 《Communications In Computational Physics》2015,18(4):1147-1180
The purpose of this study is to enhance the stability properties of our recently-developed
numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J. A. Evans, A.
Aggarwal, Y. Bazilevs, M. S. Sacks, T. J. R. Hughes, "An immersogeometric variational
framework for fluid-structure interaction: Application to bioprosthetic heart valves",
Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based
representations of shell structures into unsteady viscous incompressible flows.
In the cited work, we formulated the fluid-structure interaction (FSI) problem using
an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian
as a set of collocated constraints, at quadrature points of the surface integration
rule for the immersed interface. Because the density of quadrature points is not
controlled relative to the fluid discretization, the resulting semi-discrete problem may
be over-constrained. Semi-implicit time integration circumvents this difficulty in the
fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems
that approach steady solutions, though, we find that spatially-oscillating modes
of the Lagrange multiplier field can grow over time. In the present work, we stabilize
the semi-implicit integration scheme to prevent potential divergence of the multiplier
field as time goes to infinity. This stabilized time integration may also be applied in
pseudo-time within each time step, giving rise to a fully implicit solution method. We
discuss the theoretical implications of this stabilization scheme for several simplified
model problems, then demonstrate its practical efficacy through numerical examples. 相似文献
3.
Craig Collins Jie Shen & Steven M. Wise 《Communications In Computational Physics》2013,13(4):929-957
We present an unconditionally energy stable and uniquely solvable finite
difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised
of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model
and describes two phase very viscous flows in porous media. The scheme is based on
a convex splitting of the discrete CH energy and is semi-implicit. The equations at the
implicit time level are nonlinear, but we prove that they represent the gradient of a
strictly convex functional and are therefore uniquely solvable, regardless of time step
size. Owing to energy stability, we show that the scheme is stable in the time and space
discrete$ℓ^∞$(0,$T$;$H^1_h$) and $ℓ^2$(0,$T$;$H^2_h$) norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn-Hilliard
part, and a method based on the Vanka smoothing strategy for the Brinkman part – for
solving these equations. In particular, we provide evidence that the solver has nearly
optimal complexity in typical situations. The solver is applied to simulate spinodal
decomposition of a viscous fluid in a porous medium, as well as to the more general
problems of buoyancy- and boundary-driven flows. 相似文献
4.
A convex relaxation for the time‐optimal trajectory planning of robotic manipulators along predetermined geometric paths 下载免费PDF全文
In this paper, we deal with the problem of time‐optimal trajectory planning and feedforward controls for robotic manipulators along predetermined geometric paths. We propose a convex relaxation to generate time‐optimal trajectories and feedforward controls that are dynamically feasible with respect to the complete nonlinear dynamic model, considering both Coulomb friction and viscous friction. Even though the effects of viscous friction for time‐optimal motions become rather significant due to the required large speeds, in previous formulations, viscous friction was ignored. We present a strategic formulation that turns out non‐convex because of the consideration of viscous friction, which nonetheless leads naturally to a convex relaxation of the referred non‐convex problem. In order to numerically solve the proposed formulation, a discretization scheme is also developed. Importantly, for all the numerical instances presented in the paper, focusing on applying the algorithm results to a six‐axis industrial manipulator, the proposed convex relaxation solves exactly the original non‐convex problem. Through simulations and experimental studies on the resulting tracking errors, torque commands, and accelerometer readings for the six‐axis manipulator, we emphasize the importance of penalizing a measure of total jerk and of imposing acceleration constraints at the initial and final transitions of the trajectory. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
5.
Dominik Dierkes Florian Kummer & Dominik Plü macher 《Communications In Computational Physics》2021,30(1):288-320
We present a high-order discontinuous Galerkin (DG) scheme to solve the
system of helically symmetric Navier-Stokes equations which are discussed in [28].
In particular, we discretize the helically reduced Navier-Stokes equations emerging
from a reduction of the independent variables such that the remaining variables are: $t$, $r$, $ξ$ with $ξ=az+bϕ$, where $r$, $ϕ$, $z$ are common cylindrical coordinates and $t$ the
time. Beside this, all three velocity components are kept non-zero. A new non-singular
coordinate $η$ is introduced which ensures that a mapping of helical solutions into the
three-dimensional space is well defined. Using that, periodicity conditions for the
helical frame as well as uniqueness conditions at the centerline axis $r=0$ are derived. In
the sector near the axis of the computational domain a change of the polynomial basis
is implemented such that all physical quantities are uniquely defined at the centerline.For the temporal integration, we present a semi-explicit scheme of third order
where the full spatial operator is split into a Stokes operator which is discretized
implicitly and an operator for the nonlinear terms which is treated explicitly. Computations are conducted for a cylindrical shell, excluding the centerline axis, and for
the full cylindrical domain, where the centerline is included. In all cases we obtain the
convergence rates of order $\mathcal{O}(h^{k+1})$ that are expected from DG theory.In addition to the first DG discretization of the system of helically invariant Navier-Stokes equations, the treatment of the central axis, the resulting reduction of the DG
space, and the simultaneous use of a semi-explicit time stepper are of particular novelty. 相似文献
6.
A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh 下载免费PDF全文
Marc R. J. Charest Clinton P. T. Groth & Pierre Q. Gauthier 《Communications In Computational Physics》2015,17(3):615-656
High-order discretization techniques offer the potential to significantly reduce
the computational costs necessary to obtain accurate predictions when compared
to lower-order methods. However, efficient and universally-applicable high-order
discretizations remain somewhat illusive, especially for more arbitrary unstructured
meshes and for incompressible/low-speed flows. A novel, high-order, central essentially
non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for
the solution of the conservation equations of viscous, incompressible flows on three-dimensional
unstructured meshes. Similar to finite element methods, coordinate transformations
are used to maintain the scheme's order of accuracy even when dealing
with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied
to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes
equations and the resulting discretized equations are solved with a parallel implicit
Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach
is adopted and the resulting temporal derivatives are discretized using the family of
high-order backward difference formulas (BDF). The proposed finite-volume scheme
for fully unstructured mesh is demonstrated to provide both fast and accurate solutions
for steady and unsteady viscous flows. 相似文献
7.
Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation 下载免费PDF全文
Rongpei Zhang Xijun Yu Jiang Zhu Abimael F. D. Loula & Xia Cui 《Communications In Computational Physics》2013,14(5):1287-1303
Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe time step limits, but also take advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation. 相似文献
8.
A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations 下载免费PDF全文
Xiaodong Liu Yidong Xia Hong Luo & Lijun Xuan 《Communications In Computational Physics》2016,20(4):1016-1044
A comparative study of two classes of third-order implicit time integration
schemes is presented for a third-order hierarchical WENO reconstructed discontinuous
Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes
equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3)
scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential
algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme,
a remarkable feature of the ROW schemes is that, they only require one approximate
Jacobian matrix calculation every time step, thus considerably reducing the overall
computational cost. A variety of test cases, ranging from inviscid flows to DNS of
turbulent flows, are presented to assess the performance of these schemes. Numerical
experiments demonstrate that the third-order ROW scheme for the DAEs of index-2
can not only achieve the designed formal order of temporal convergence accuracy in
a benchmark test, but also require significantly less computing time than its ESDIRK3
counterpart to converge to the same level of discretization errors in all of the flow
simulations in this study, indicating that the ROW methods provide an attractive alternative
for the higher-order time-accurate integration of the unsteady compressible
Navier-Stokes equations. 相似文献
9.
High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes 下载免费PDF全文
Weijie Zhang Yulong Xing Yinhua Xia & Yan Xu 《Communications In Computational Physics》2022,31(3):771-815
In this paper, we propose a high-order accurate discontinuous Galerkin
(DG) method for the compressible Euler equations under gravitational fields on unstructured meshes. The scheme preserves a general hydrostatic equilibrium state and
provably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitation
on rectangular meshes, the extension to triangular meshes is conceptually plausible
but highly nontrivial. We first introduce a special way to recover the equilibrium state
and then design a group of novel variables at the interface of two adjacent cells, which
plays an important role in the well-balanced and positivity-preserving properties. One
main challenge is that the well-balanced schemes may not have the weak positivity
property. In order to achieve the well-balanced and positivity-preserving properties
simultaneously while maintaining high-order accuracy, we carefully design DG spatial discretization with well-balanced numerical fluxes and suitable source term approximation. For the ideal gas, we prove that the resulting well-balanced scheme, coupled with strong stability preserving time discretizations, satisfies a weak positivity
property. A simple existing limiter can be applied to enforce the positivity-preserving
property, without losing high-order accuracy and conservation. Extensive one- and
two-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness. 相似文献
10.
Comparison of Fifth-Order WENO Scheme and Finite Volume WENO-Gas-Kinetic Scheme for Inviscid and Viscous Flow Simulation 下载免费PDF全文
The development of high-order schemes has been mostly concentrated on
the limiters and high-order reconstruction techniques. In this paper, the effect of the
flux functions on the performance of high-order schemes will be studied. Based on the
same WENO reconstruction, two schemes with different flux functions, i.e., the fifth-order WENO method and the WENO-Gas-Kinetic scheme (WENO-GKS), will be compared. The fifth-order finite difference WENO-SW scheme is a characteristic variable
reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms, the sixth-order central difference for viscous terms, and three stages Runge-Kutta time stepping for the time integration. On the other hand, the finite volume
WENO-GKS is a conservative variable reconstruction based method with the same
WENO reconstruction. But it evaluates a time dependent gas distribution function
along a cell interface, and updates the flow variables inside each control volume by
integrating the flux function along the boundary of the control volume in both space
and time. In order to validate the robustness and accuracy of the schemes, both methods are tested under a wide range of flow conditions: vortex propagation, Mach 3
step problem, and the cavity flow at Reynolds number 3200. Our study shows that
both WENO-SW and WENO-GKS yield quantitatively similar results and agree with
each other very well provided a sufficient grid resolution is used. With the reduction of mesh points, the WENO-GKS behaves to have less numerical dissipation and
present more accurate solutions than those from the WENO-SW in all test cases. For
the Navier-Stokes equations, since the WENO-GKS couples inviscid and viscous terms
in a single flux evaluation, and the WENO-SW uses an operator splitting technique, it
appears that the WENO-SW is more sensitive to the WENO reconstruction and boundary treatment. In terms of efficiency, the finite volume WENO-GKS is about 4 times
slower than the finite difference WENO-SW in two dimensional simulations. The current study clearly shows that besides high-order reconstruction, an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions. In a physical flow, the transport, stress deformation, heat conduction, and viscous heating are all coupled in a single gas evolution process. Therefore,
it is preferred to develop such a scheme with multi-dimensionality, and unified treatment of inviscid and dissipative terms. A high-order scheme does prefer a high-order
gas evolution model. Even with the rapid advances of high-order reconstruction techniques, the first-order dynamics of the Riemann solution becomes the bottleneck for
the further development of high-order schemes. In order to avoid the weakness of the
low order flux function, the development of high-order schemes relies heavily on the
weak solution of the original governing equations for the update of additional degree
of freedom, such as the non-conservative gradients of flow variables, which cannot be
physically valid in discontinuous regions. 相似文献
11.
Our aim in this article is to improve the understanding of the colocated finite
volume schemes for the incompressible Navier-Stokes equations. When all the variables are colocated, that means here when the velocities and the pressure are computed
at the same place (at the centers of the control volumes), these unknowns must be properly coupled. Consequently, the choice of the time discretization and the method used
to interpolate the fluxes at the edges of the control volumes are essentials. In the first
and second parts of this article, two different time discretization schemes are considered with a colocated space discretization and we explain how the unknowns can be
correctly coupled. Numerical simulations are presented in the last part of the article.
This paper is not a comparison between staggered grid schemes and colocated schemes
(for this, see, e.g., [15, 22]). We plan, in the future, to use a colocated space discretization and the multilevel method of [4] initially applied to the two dimensional Burgers
problem, in order to solve the incompressible Navier-Stokes equations. One advantage
of colocated schemes is that all variables share the same location, hence, the possibility to use hierarchical space discretizations more easily when multilevel methods are
used. For this reason, we think that it is important to study this family of schemes. 相似文献
12.
Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws 下载免费PDF全文
Andreas Hiltebrand & Siddhartha Mishra 《Communications In Computational Physics》2015,17(5):1320-1359
An entropy stable fully discrete shock capturing space-time Discontinuous
Galerkin (DG) method was proposed in a recent paper [20] to approximate hyperbolic
systems of conservation laws. This numerical scheme involves the solution of a
very large nonlinear system of algebraic equations, by a Newton-Krylov method, at
every time step. In this paper, we design efficient preconditioners for the large, nonsymmetric
linear system, that needs to be solved at every Newton step. Two sets of
preconditioners, one of the block Jacobi and another of the block Gauss-Seidel type are
designed. Fourier analysis of the preconditioners reveals their robustness and a large
number of numerical experiments are presented to illustrate the gain in efficiency that
results from preconditioning. The resulting method is employed to compute approximate
solutions of the compressible Euler equations, even for very high CFL numbers. 相似文献
13.
New Splitting Methods for Convection-Dominated Diffusion Problems and Navier-Stokes Equations 下载免费PDF全文
Feng Shi Guoping Liang Yubo Zhao & Jun Zou 《Communications In Computational Physics》2014,16(5):1239-1262
We present a new splitting method for time-dependent convention-dominated diffusion problems. The original convention diffusion system is split into two
sub-systems: a pure convection system and a diffusion system. At each time step, a
convection problem and a diffusion problem are solved successively. A few important features of the scheme lie in the facts that the convection subproblem is solved
explicitly and multistep techniques can be used to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; while
the diffusion subproblem is always self-adjoint and coercive so that they can be solved
efficiently using many existing optimal preconditioned iterative solvers. The scheme
can be extended for solving the Navier-Stokes equations, where the nonlinearity is
resolved by a linear explicit multistep scheme at the convection step, while only a generalized Stokes problem is needed to solve at the diffusion step and the major stiffness
matrix stays invariant in the time marching process. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step
and multistep variants of the new scheme. 相似文献
14.
A Fourth-Order Upwinding Embedded Boundary Method (UEBM) for Maxwell's Equations in Media with Material Interfaces: Part I 下载免费PDF全文
In this paper, we present a new fourth-order upwinding embedded boundary method (UEBM) over Cartesian grids, originally proposed in the Journal of Computational Physics [190 (2003), pp. 159-183.] as a second-order method for treating material interfaces for Maxwell's equations. In addition to the idea of the UEBM to evolve solutions at interfaces, we utilize the ghost fluid method to construct finite difference approximation of spatial derivatives at Cartesian grid points near the material interfaces. As a result, Runge-Kutta type time discretization can be used for the semidiscretized system to yield an overall fourth-order method, in contrast to the original second-order UEBM based on a Lax-Wendroff type difference. The final scheme allows time step sizes independent of the interface locations. Numerical examples are given to demonstrate the fourth-order accuracy as well as the stability of the method. We tested the scheme for several wave problems with various material interface locations, including electromagnetic scattering of a plane wave incident on a planar boundary and a two-dimensional electromagnetic application with an interface parallel to the y-axis. 相似文献
15.
Higher-Order Compact Scheme for the Incompressible Navier-Stokes Equations in Spherical Geometry 下载免费PDF全文
T. V. S. Sekhar B. Hema Sundar Raju & Y. V. S. S. Sanyasiraju 《Communications In Computational Physics》2012,11(1):99-113
A higher-order compact scheme on the nine point 2-D stencil is developed
for the steady stream-function vorticity form of the incompressible Navier-Stokes (NS) equations in spherical polar coordinates, which was used earlier only for the cartesian and cylindrical geometries. The steady, incompressible, viscous and axially symmetric flow past a sphere is used as a model problem. The non-linearity in the N-S
equations is handled in a comprehensive manner avoiding complications in calculations. The scheme is combined with the multigrid method to enhance the convergence
rate. The solutions are obtained over a non-uniform grid generated using the transformation r = eξ while maintaining a uniform grid in the computational plane. The
superiority of the higher order compact scheme is clearly illustrated in comparison
with upwind scheme and defect correction technique at high Reynolds numbers by
taking a large domain. This is a pioneering effort, because for the first time, the fourth
order accurate solutions for the problem of viscous flow past a sphere are presented
here. The drag coefficient and surface pressures are calculated and compared with
available experimental and theoretical results. It is observed that these values simulated over coarser grids using the present scheme are more accurate when compared to
other conventional schemes. It has also been observed that the flow separation initially
occurred at Re=21. 相似文献
16.
A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws 下载免费PDF全文
Rapha& euml l Loub& egrave re Michael Dumbser & Steven Diot 《Communications In Computational Physics》2014,16(3):718-763
In this paper, we investigate the coupling of the Multi-dimensional Optimal
Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER)
approach in order to design a new high order accurate, robust and computationally
efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and
three space dimensions, respectively. The Multi-dimensional Optimal Order Detection
(MOOD) method for 2D and 3D geometries has been introduced in a recent series of
papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite
Volume scheme in space, using polynomial reconstructions with a posteriori detection
and polynomial degree decrementing processes to deal with shock waves and other
discontinuities. In the following work, the time discretization is performed with an
elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached
on smooth solutions, while spurious oscillations near singularities are prevented. The
ADER technique not only reduces the cost of the overall scheme as shown
on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO
schemes and the new ADER-MOOD approach has been carried out for high-order
schemes in space and time in terms of cost, robustness, accuracy and efficiency. The
main finding of this paper is that the combination of ADER with MOOD generally
outperforms the one of ADER and WENO either because at given accuracy MOOD isless expensive (memory and/or CPU time), or because it is more accurate for a given
grid resolution. A large suite of classical numerical test problems has been solved
on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations
of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations
(RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic
partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements. 相似文献
17.
A High Order Sharp-Interface Method with Local Time Stepping for Compressible Multiphase Flows 下载免费PDF全文
Angela Ferrari Claus-Dieter Munz & Bernhard Weigand 《Communications In Computational Physics》2011,9(1):205-230
In this paper, a new sharp-interface approach to simulate compressible
multiphase flows is proposed. The new scheme consists of a high order WENO finite volume scheme for solving the Euler equations coupled with a high order path-conservative
discontinuous Galerkin finite element scheme to evolve an indicator function
that tracks the material interface. At the interface our method applies ghost cells
to compute the numerical flux, as the ghost fluid method. However, unlike the original
ghost fluid scheme of Fedkiw et al. [15], the state of the ghost fluid is derived
from an approximate-state Riemann solver, similar to the approach proposed in [25],
but based on a much simpler formulation. Our formulation leads only to one single
scalar nonlinear algebraic equation that has to be solved at the interface, instead of
the system used in [25]. Away from the interface, we use the new general Osher-type
flux recently proposed by Dumbser and Toro [13], which is a simple but complete Riemann
solver, applicable to general hyperbolic conservation laws. The time integration
is performed using a fully-discrete one-step scheme, based on the approaches recently
proposed in [5, 7]. This allows us to evolve the system also with time-accurate local
time stepping. Due to the sub-cell resolution and the subsequent more restrictive
time-step constraint of the DG scheme, a local evolution for the indicator function is
applied, which is matched with the finite volume scheme for the solution of the Euler
equations that runs with a larger time step. The use of a locally optimal time step
avoids the introduction of excessive numerical diffusion in the finite volume scheme.
Two different fluids have been used, namely an ideal gas and a weakly compressible
fluid modeled by the Tait equation. Several tests have been computed to assess the
accuracy and the performance of the new high order scheme. A verification of our
algorithm has been carefully carried out using exact solutions as well as a comparison
with other numerical reference solutions. The material interface is resolved sharply
and accurately without spurious oscillations in the pressure field. 相似文献
18.
An all speed scheme for the Isentropic Euler equations is presented in this
paper. When the Mach number tends to zero, the compressible Euler equations converge
to their incompressible counterpart, in which the density becomes a constant. Increasing
approximation errors and severe stability constraints are the main difficulty
in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit
time discretization, in which the low Mach number stiff term is divided into
two parts, one being treated explicitly and the other one implicitly. Moreover, the flux
of the density equation is also treated implicitly and an elliptic type equation is derived
to obtain the density. In this way, the correct limit can be captured without requesting
the mesh size and time step to be smaller than the Mach number. Compared with
previous semi-implicit methods [11,13,29], firstly, nonphysical oscillations can be suppressed
by choosing proper parameter, besides, only a linear elliptic equation needs to
be solved implicitly which reduces much computational cost. We develop this semi-implicit
time discretization in the framework of a first order Local Lax-Friedrichs (or
Rusanov) scheme and numerical tests are displayed to demonstrate its performances. 相似文献
19.
This paper is motivated by an optimal boundary control problem for the cooling process of molten and already formed glass down to room temperature. The high temperatures at which glass is processed demand to include radiative heat transfer in the computational model. Since the complete radiative heat transfer equations are too complex for optimization purposes, we use simplified approximations of spherical harmonics coupled with a practically relevant frequency bands model. The optimal control problem is considered as a partial differential algebraic equation (PDAE)‐constrained optimization problem with box constraints on the control. In this paper, we augment the objective by a functional depending on the state gradient, which forces a minimization of thermal stress inside the glass. To guarantee consistent and grid‐independent values of the reduced objective gradient at the end of the cooling process, we pursue two approaches. The first includes the temperature gradient with a time‐dependent linearly decreasing weight. In the second approach, we augment the objective functional by the final state tracking and final state gradient optimization. To determine an optimal boundary control, we apply a projected gradient method with the Armijo step size rule. The reduced objective gradient is computed by the continuous adjoint approach. The arising time‐dependent PDAEs are numerically solved by variable step size one‐step methods of Rosenbrock type in time and adaptive multilevel finite elements in space. We present two‐dimensional numerical results for an infinitely long glass block and compare the two different approaches derived to ensure consistency at the end of the cooling process. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
20.
This paper presents a new finite-volume discretization of a generalised Lattice Boltzmann equation (LBE) on unstructured grids. This equation is the continuum LBE, with the addition of a second order time derivative term (memory), and is derived from a second-order differential form of the semi-discrete Boltzmann equation in its implicit form. The new scheme, named unstructured lattice Boltzmann equation with memory (ULBEM), can be advanced in time with a larger time-step than the previous unstructured LB formulations, and a theoretical demonstration of the improved stability is provided. Taylor vortex simulations show that the viscosity is the same as with standard ULBE and demonstrates that the new scheme improves both stability and accuracy. Model validation is also demonstrated by simulating backward-facing step flow at low and moderate Reynolds numbers, as well as by comparing the reattachment length of the recirculating eddy behind the step against experimental and numerical data available in literature. 相似文献