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1.
Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme
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Louisa Schlachter & Claudia Totzeck 《Communications In Computational Physics》2020,28(4):1585-1608
We study an identification problem which estimates the parameters of the
underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin
method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain
initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on
identifying the correct endpoints of this interval. The first-order optimality conditions
from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward
schemes with the Runge-Kutta method. To illustrate the feasibility of the approach,
we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parameters
of the random variable in the uncertain differential equation even if discontinuities are
present. 相似文献
2.
Polynomial chaos methods (and generalized polynomial chaos methods) have been extensively applied to analyze PDEs that contain uncertainties. However, this approach is rarely applied to hyperbolic systems. In this paper we analyze the properties of the resulting deterministic system of equations obtained by stochastic Galerkin projection. We consider a simple model of a scalar wave equation with random wave speed. We show that when uncertainty causes the change of characteristic directions, the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues. A consistent method of imposing the boundary conditions is proposed and its convergence is established. Numerical examples are presented to support the analysis. 相似文献
3.
Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws
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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
4.
This paper is concerned with a new version of the Osher-Solomon Riemann
solver and is based on a numerical integration of the path-dependent dissipation matrix.
The resulting scheme is much simpler than the original one and is applicable to
general hyperbolic conservation laws, while retaining the attractive features of the original
solver: the method is entropy-satisfying, differentiable and complete in the sense
that it attributes a different numerical viscosity to each characteristic field, in particular
to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system
is used. To illustrate the potential of the proposed scheme we show applications
to the following hyperbolic conservation laws: Euler equations of compressible gasdynamics
with ideal gas and real gas equation of state, classical and relativistic MHD
equations as well as the equations of nonlinear elasticity. To the knowledge of the authors,
apart from the Euler equations with ideal gas, an Osher-type scheme has never
been devised before for any of these complicated PDE systems. Since our new general
Riemann solver can be directly used as a building block of high order finite volume
and discontinuous Galerkin schemes we also show the extension to higher order of
accuracy and multiple space dimensions in the new framework of PNPM schemes on
unstructured meshes recently proposed in [9]. 相似文献
5.
Shuangzhang Tu Gordon W. Skelton & Qing Pang 《Communications In Computational Physics》2011,9(2):441-480
This paper presents a novel high-order space-time method for hyperbolic
conservation laws. Two important concepts, the staggered space-time mesh of the
space-time conservation element/solution element (CE/SE) method and the local discontinuous
basis functions of the space-time discontinuous Galerkin (DG) finite element
method, are the two key ingredients of the new scheme. The staggered space-time
mesh is constructed using the cell-vertex structure of the underlying spatial mesh.
The universal definitions of CEs and SEs are independent of the underlying spatial
mesh and thus suitable for arbitrarily unstructured meshes. The solution within each
physical time step is updated alternately at the cell level and the vertex level. For
this solution updating strategy and the DG ingredient, the new scheme here is termed
as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy
is achieved by employing high-order Taylor polynomials as the basis functions
inside each SE. The present DG-CVS exhibits many advantageous features such as
Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease
of handling boundary conditions. Several numerical tests including the scalar advection
equations and compressible Euler equations will demonstrate the performance of
the new method. 相似文献
6.
A Weighted Runge-Kutta Discontinuous Galerkin Method for 3D Acoustic and Elastic Wave-Field Modeling
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Numerically solving 3D seismic wave equations is a key requirement for
forward modeling and inversion. Here, we propose a weighted Runge-Kutta discontinuous Galerkin (WRKDG) method for 3D acoustic and elastic wave-field modeling. For this method, the second-order seismic wave equations in 3D heterogeneous anisotropic media are transformed into a first-order hyperbolic system, and
then we use a discontinuous Galerkin (DG) solver based on numerical-flux formulations for spatial discretization. The time discretization is based on an implicit diagonal Runge-Kutta (RK) method and an explicit iterative technique, which avoids
solving a large-scale system of linear equations. In the iterative process, we introduce
a weighting factor. We investigate the numerical stability criteria of the 3D method in
detail for linear and quadratic spatial basis functions. We also present a 3D analysis of
numerical dispersion for the full discrete approximation of acoustic equation, which
demonstrates that the WRKDG method can efficiently suppress numerical dispersion
on coarse grids. Numerical results for several different 3D models including homogeneous and heterogeneous media with isotropic and anisotropic cases show that the 3D
WRKDG method can effectively suppress numerical dispersion and provide accurate
wave-field information on coarse mesh. 相似文献
7.
Yongping Cheng Haiyun Dong Maojun Li & Weizhi Xian 《Communications In Computational Physics》2020,28(4):1437-1463
In this paper, we focus on the numerical simulation of the two-layer shallow water equations over variable bottom topography. Although the existing numerical schemes for the single-layer shallow water equations can be extended to two-layer
shallow water equations, it is not a trivial work due to the complexity of the equations.
To achieve the well-balanced property of the numerical scheme easily, the two-layer
shallow water equations are reformulated into a new form by introducing two auxiliary variables. Since the new equations are only conditionally hyperbolic and their
eigenstructure cannot be easily obtained, we consider the utilization of the central discontinuous Galerkin method which is free of Riemann solvers. By choosing the values
of the auxiliary variables suitably, we can prove that the scheme can exactly preserve
the still-water solution, and thus it is a truly well-balanced scheme. To ensure the
non-negativity of the water depth, a positivity-preserving limiter and a special approximation to the bottom topography are employed. The accuracy and validity of the
numerical method will be illustrated through some numerical tests. 相似文献
8.
A Sparse Grid Discrete Ordinate Discontinuous Galerkin Method for the Radiative Transfer Equation
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Jianguo Huang & Yue Yu 《Communications In Computational Physics》2021,30(4):1009-1036
The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case,
leading to great difficulties in numerical simulation. To tackle this bottleneck, we first
use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we
make the spatial discretization by means of the discontinuous Galerkin (DG) method
combined with the sparse grid method. The final linear system is solved by the block
Gauss-Seidal iteration method. The computational complexity and error analysis are
developed in detail, which show the new method is more efficient than the original
discrete ordinate DG method. A series of numerical results are performed to validate
the convergence behavior and effectiveness of the proposed method. 相似文献
9.
Seshu Tirupathi Jan S. Hesthaven & Yan Liang 《Communications In Computational Physics》2015,18(1):230-246
Discontinuous Galerkin (DG) and matrix-free finite element methods with
a novel projective pressure estimation are combined to enable the numerical modeling
of magma dynamics in 2D and 3D using the library deal.II. The physical model
is an advection-reaction type system consisting of two hyperbolic equations to evolve
porosity and soluble mineral abundance at local chemical equilibrium and one elliptic
equation to recover global pressure. A combination of a discontinuous Galerkin
method for the advection equations and a finite element method for the elliptic equation
provide a robust and efficient solution to the channel regime problems of the
physical system in 3D. A projective and adaptively applied pressure estimation is employed
to significantly reduce the computational wall time without impacting the overall
physical reliability in the modeling of important features of melt segregation, such
as melt channel bifurcation in 2D and 3D time dependent simulations. 相似文献
10.
An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficient
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Zhiwen Zhang Xin Hu Thomas Y. Hou Guang Lin & Mike Yan 《Communications In Computational Physics》2014,16(3):571-598
In this paper, we present an adaptive, analysis of variance (ANOVA)-based
data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven
stochastic method. To handle high-dimensional stochastic problems, we investigate
the use of adaptive ANOVA decomposition in the stochastic space as an effective
dimension-reduction technique. To improve the slow convergence of the generalized
polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the
data-driven stochastic method (DSM) for speed up. An essential ingredient of the
DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary
conditions.Our ANOVA-DSM consists of offline and online stages. In the offline stage, the
original high-dimensional stochastic problem is decomposed into a series of low-dimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using
the Karhunen-Loève expansion (KLE) and a two-level preconditioning optimization
approach. Multiple trial functions are used to enrich the stochastic basis and improve
the accuracy. In the online stage, we solve each stochastic subproblem for any given
forcing function by projecting the stochastic solution into the data-driven stochastic
basis constructed offline. In our ANOVA-DSM framework, solving the original high-dimensional stochastic problem is reduced to solving a series of ANOVA-decomposed
stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided
to further reduce the number of the stochastic subproblems and speed up our method.
To demonstrate the accuracy and efficiency of our method, numerical examples are
presented for one- and two-dimensional elliptic PDEs with random coefficients. 相似文献
11.
Alan R. Schiemenz Marc A. Hesse & Jan S. Hesthaven 《Communications In Computational Physics》2011,10(2):433-452
A high-order discretization consisting of a tensor product of the Fourier collocation
and discontinuous Galerkin methods is presented for numerical modeling of
magma dynamics. The physical model is an advection-reaction type system consisting
of two hyperbolic equations and one elliptic equation. The high-order solution
basis allows for accurate and efficient representation of compaction-dissolution waves
that are predicted from linear theory. The discontinuous Galerkin method provides
a robust and efficient solution to the eigenvalue problem formed by linear stability
analysis of the physical system. New insights into the processes of melt generation
and segregation, such as melt channel bifurcation, are revealed from two-dimensional
time-dependent simulations. 相似文献
12.
A Runge Kutta Discontinuous Galerkin Method for Lagrangian Compressible Euler Equations in Two-Dimensions
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Zhenzhen Li Xijun Yu Jiang Zhu & Zupeng Jia 《Communications In Computational Physics》2014,15(4):1184-1206
This paper presents a new Lagrangian type scheme for solving the Euler
equations of compressible gas dynamics. In this new scheme the system of equations
is discretized by Runge-Kutta Discontinuous Galerkin (RKDG) method, and the mesh
moves with the fluid flow. The scheme is conservative for the mass, momentum and
total energy and maintains second-order accuracy. The scheme avoids solving the geometrical
part and has free parameters. Results of some numerical tests are presented
to demonstrate the accuracy and the non-oscillatory property of the scheme. 相似文献
13.
Tom Lefebvre Frederik De Belie Guillaume Crevecoeur 《Optimal control applications & methods.》2020,41(3):833-848
We propose a framework tailored to robust optimal control (OC) problems subject to parametric model uncertainty of system dynamics. First, we identify a generic class of robust objective kernels that are based on the class of deterministic quadratic objectives. It is demonstrated how such kernels can be expressed as a function of the stochastic moments of the state and how the objective terms relate to the robustness and performance of the optimal solution. Second, we engage the generalized polynomial chaos (gPC) framework to propagate uncertainty along the state trajectory. Integrating both frameworks makes way to reformulate the problem as a deterministic OC problem in function of the gPC expansion coefficients that can be solved using existing methods. We apply the framework to solve the problem of robust optimal startup behavior of a nonlinear mechanical drivetrain that is subject to a bifurcation in its dynamics. 相似文献
14.
A fully discrete discontinuous Galerkin method is introduced for solving
time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in
our scheme, discontinuous Galerkin methods are used to discretize not only the spatial
domain but also the temporal domain. The proposed numerical scheme is proved to be
unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and
spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in
time step is observed numerically for the numerical fluxes w.r.t. temporal variable at
the grid points. 相似文献
15.
Rajesh Gandham David Medina & Timothy Warburton 《Communications In Computational Physics》2015,18(1):37-64
We discuss the development, verification, and performance of a GPU accelerated
discontinuous Galerkin method for the solutions of two dimensional nonlinear
shallow water equations. The shallow water equations are hyperbolic partial differential
equations and are widely used in the simulation of tsunami wave propagations.
Our algorithms are tailored to take advantage of the single instruction multiple data
(SIMD) architecture of graphic processing units. The time integration is accelerated by
local time stepping based on a multi-rate Adams-Bashforth scheme. A total variational
bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter
is coupled with a mass and momentum conserving positivity preserving limiter
for the special treatment of a dry or partially wet element in the triangulation. Accuracy,
robustness and performance are demonstrated with the aid of test cases. Furthermore,
we developed a unified multi-threading model OCCA. The kernels expressed
in OCCA model can be cross-compiled with multi-threading models OpenCL, CUDA,
and OpenMP. We compare the performance of the OCCA kernels when cross-compiled
with these models. 相似文献
16.
A High Order Sharp-Interface Method with Local Time Stepping for Compressible Multiphase Flows
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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
17.
We present a parallel Schwarz type domain decomposition preconditioned
recycling Krylov subspace method for the numerical solution of stochastic indefinite
elliptic equations with two random coefficients. Karhunen-Loève expansions are used
to represent the stochastic variables and the stochastic Galerkin method with double
orthogonal polynomials is used to derive a sequence of uncoupled deterministic
equations. We show numerically that the Schwarz preconditioned recycling GMRES
method is an effective technique for solving the entire family of linear systems and, in
particular, the use of recycled Krylov subspaces is the key element of this successful
approach. 相似文献
18.
John Loverich Ammar Hakim & Uri Shumlak 《Communications In Computational Physics》2011,9(2):240-268
A discontinuous Galerkin method for the ideal 5 moment two-fluid plasma
system is presented. The method uses a second or third order discontinuous Galerkin
spatial discretization and a third order TVD Runge-Kutta time stepping scheme. The
method is benchmarked against an analytic solution of a dispersive electron acoustic
square pulse as well as the two-fluid electromagnetic shock [1] and existing numerical
solutions to the GEM challenge magnetic reconnection problem [2]. The algorithm can
be generalized to arbitrary geometries and three dimensions. An approach to maintaining
small gauge errors based on error propagation is suggested. 相似文献
19.
An LDG Method for Stochastic Cahn-Hilliard Type Equation Driven by General Multiplicative Noise Involving Second-Order Derivative
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Li Zhou & Yunzhang Li 《Communications In Computational Physics》2022,31(2):516-547
In this paper, we propose a local discontinuous Galerkin (LDG) method for
the multi-dimensional stochastic Cahn-Hilliard type equation in a general form, which
involves second-order derivative $∆u$ in the multiplicative noise. The stability of our
scheme is proved for arbitrary polygonal domain with triangular meshes. We get the
sub-optimal error estimate $\mathbb{O}(h^k)$ if the Cartesian meshes with $Q^k$ elements are used.
Numerical examples are given to display the performance of the LDG method. 相似文献
20.
Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes
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Walter Boscheri Michael Dumbser & Elena Gaburro 《Communications In Computational Physics》2022,32(1):259-298
We propose a new high order accurate nodal discontinuous Galerkin (DG)
method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical
polynomials of degree $N$ inside each element, in our new approach the discrete solution
is represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon.
We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG
method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each
sub-triangle are defined, as usual, on a universal reference element, hence allowing to
compute universal mass, flux and stiffness matrices for the subgrid triangles once and
for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured
nature of the computational grid. High order of accuracy in time is achieved thanks
to the ADER approach, making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results
have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained
by the corresponding modal DG version of the scheme. 相似文献