Galerkin Methods For Hyperbolic Equations Research Articles

A Deep Learning Based Discontinuous Galerkin Method for Hyperbolic Equations with Discontinuous Solutions and Random Uncertainties

A Deep Learning Based Discontinuous Galerkin Method for Hyperbolic Equations with Discontinuous Solutions and Random Uncertainties

Arbitrary Lagrangian--Eulerian Discontinuous Galerkin Method for Hyperbolic Equations Involving $\delta$-Singularities

In this paper, we develop and analyze an arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) method for solving one-dimensional hyperbolic equations involving $\delta$-singularities on moving meshes. The $L^2$ and negative norm error estimates are proven for the ALE-DG approximation. More precisely, when choosing the approximation space with piecewise $k$th degree polynomials, the convergence rate in $L^2$-norm for the scheme with the upwind numerical flux is $(k+1)$th order in the region apart from the singularities, the convergence rate in $H^{-(k+1)}$ norm for the scheme with the monotone fluxes in the whole domain is $k$th order, the convergence rate in $H^{-(k+2)}$ norm for the scheme with the upwind flux in the whole domain can achieve $(k+\frac{1}{2})$th order, and the convergence rate in $H^{-(k+1)}(R\backslash R_T)$ norm for the scheme with the upwind flux is $(2k+1)$th order, where $R_T$ is the pollution region at time $T$ due to the singularities. Moreover, numerically the $(2k+1)$th order accuracy for the postprocessed solution in the smooth region can be obtained, which is produced by convolving the ALE-DG solution with a suitable kernel consisting of B-splines. Numerical examples are shown to demonstrate the accuracy and capability of the ALE-DG method for the hyperbolic equations involving $\delta$-singularity on moving meshes.

Filtered stochastic Galerkin methods for hyperbolic equations

Uncertainty Quantification for nonlinear hyperbolic problems becomes a challenging task in the vicinity of shocks. Standard intrusive methods, such as Stochastic Galerkin (SG), lead to oscillatory solutions and can result in non-hyperbolic moment systems. The intrusive polynomial moment (IPM) method guarantees hyperbolicity but comes at higher numerical costs. In this paper, we filter the generalized polynomial chaos (gPC) coefficients of the SG approximation, which allows a numerically cheap reduction of oscillations. The derived filter is based on Lasso regression which sets small gPC coefficients of high order to zero. We adaptively and automatically choose the filter strength to obtain a zero-valued highest order moment. The filtered SG method is tested for Burgers' and the Euler equations. Results show a reduction of oscillations at shocks, which leads to an improved approximation of expectation values and the variance compared to SG and IPM.

The Discrete Stochastic Galerkin Method for Hyperbolic Equations with Non-smooth and Random Coefficients

We develop a general polynomial chaos (gPC) based stochastic Galerkin (SG) for hyperbolic equations with random and singular coefficients. Due to the singular nature of the solution, the standard gPC-SG methods may suffer from a poor or even non convergence. Taking advantage of the fact that the discrete solution, by the central type finite difference or finite volume approximations in space and time for example, is smoother, we first discretize the equation by a smooth finite difference or finite volume scheme, and then use the gPC-SG approximation to the discrete system. The jump condition at the interface is treated using the immersed upwind methods introduced in Jin (Proc Symp Appl Math 67(1):93–104, 2009) and Jin and Wen (Commun Math Sci 3:285–315, 2005). This yields a method that converges with the spectral accuracy for finite mesh size and time step. We use a linear hyperbolic equation with discontinuous and random coefficient, and the Liouville equation with discontinuous and random potential, to illustrate our idea, with both one and second order spatial discretizations. Spectral convergence is established for the first equation, and numerical examples for both equations show the desired accuracy of the method.

Discontinuous Galerkin method for hyperbolic equations involving $$\delta $$ -singularities: negative-order norm error estimates and applications

In this paper, we develop and analyze discontinuous Galerkin (DG) methods to solve hyperbolic equations involving $$\delta $$ -singularities. Negative-order norm error estimates for the accuracy of DG approximations to $$\delta $$ -singularities are investigated. We first consider linear hyperbolic conservation laws in one space dimension with singular initial data. We prove that, by using piecewise $$k$$ th degree polynomials, at time $$t$$ , the error in the $$H^{-(k+2)}$$ norm over the whole domain is $$(k+1/2)$$ th order, and the error in the $$H^{-(k+1)}(\mathbb R \backslash \mathcal R _t)$$ norm is $$(2k+1)$$ th order, where $$\mathcal R _t$$ is the pollution region due to the initial singularity with the width of order $$\mathcal O (h^{1/2} \log (1/h))$$ and $$h$$ is the maximum cell length. As an application of the negative-order norm error estimates, we convolve the numerical solution with a suitable kernel which is a linear combination of B-splines, to obtain $$L^2$$ error estimate of $$(2k+1)$$ th order for the post-processed solution. Moreover, we also obtain high order superconvergence error estimates for linear hyperbolic conservation laws with singular source terms by applying Duhamel's principle. Numerical examples including an acoustic equation and the nonlinear rendez-vous algorithms are given to demonstrate the good performance of DG methods for solving hyperbolic equations involving $$\delta $$ -singularities.

On the order of convergence of the discontinuous Galerkin method for hyperbolic equations

The basic error estimate for the discontinuous Galerkin method for hyperbolic equations indicates an O(h n+½ ) convergence rate for nth degree polynomial approximation over a triangular mesh of size h. However, the optimal O(h +1 ) rate is frequently seen in practice. Here we extend the class of meshes for which sharpness of the O(h +½ ) estimate can be demonstrated, using as an example a problem with a nonaligned mesh in which all triangle sides are bounded away from the characteristic direction. The key to realizing h n+ ½ convergence is a mesh which, to the extent possible, directs the error to lower frequency modes which are approximated, not damped, as h→ 0.

On the justification of the Galerkin method for hyperbolic equations

On the justification of the Galerkin method for hyperbolic equations

High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere

High-order triangle-based discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite element-type area integrals are evaluated using order 2 N Gauss cubature rules. This leads to a full mass matrix which, unlike for continuous Galerkin (CG) methods such as the spectral element (SE) method presented in Giraldo and Warburton [A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys. 207 (2005) 129–150], is small, local and efficient to invert. Two types of finite volume-type flux integrals are studied: a set based on Gauss–Lobatto quadrature points (order 2 N − 1) and a set based on Gauss quadrature points (order 2 N). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with 2 N integration being the most accurate of the four DG methods studied. The strong advection form with 2 N integration performed extremely well even for flows with shock waves. The strong conservation form with 2 N − 1 integration yielded results almost as good as those with 2 N while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix.

On error estimates in the Galerkin method for hyperbolic equations

We consider the Cauchy problem in a Hilbert space for a second-order abstract quasilinear hyperbolic equation with variable operator coefficients and nonsmooth (but Bochner integrable) free term. For this problem, we establish an a priori energy error estimate for the semidiscrete Galerkin method with an arbitrary choice of projection subspaces. Also, we establish some results on existence and uniqueness of an exact weak solution. We give an explicit error estimate for the finite element method and the Galerkin method in Mikhlin form.

A WAVELET-TAYLOR GALERKIN METHOD FOR PARABOLIC AND HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this study a set of new space and time accurate numerical methods based on different time marching schemes such as Euler, leap-frog and Crank-Nicolson for partial differential equations of the form [Formula: see text], where ℒ is linear differential operator and [Formula: see text] is a nonlinear function, are proposed. To produce accurate temporal differencing, the method employs forward/backward time Taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation. This yields a generalized time discretized scheme which is approximated in space by Galerkin method. The compactly supported orthogonal wavelet bases developed by Daubechies are used in Galerkin scheme. This new wavelet-Taylor Galerkin approach is successively applied to heat equation, convection equation and inviscid Burgers' equation.

Extension of a Post Processing Technique for the Discontinuous Galerkin Method for Hyperbolic Equations with Application to an Aeroacoustic Problem

In this paper we further explore a local postprocessing technique, originally developed by Bramble and Schatz [Math. Comp., 31 (1977), pp. 94--111] using continuous finite element methods for elliptic problems and later by Cockburn et al. [Math. Comp., 72 (2003), pp. 577--606] using discontinuous Galerkin methods for hyperbolic equations. We investigate the technique in the context of superconvergence of the derivatives of the numerical solution, two space dimensions for both tensor product local basis and the usual kth degree polynomials basis, multidomain problems with different mesh sizes, variable coefficient linear problems including those with discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. We demonstrate through extensive numerical examples that the technique is very effective in all these situations in enhancing the accuracy of the discontinuous Galerkin solutions.

Stabilized discontinuous Galerkin method for hyperbolic equations

In this work a new stabilization technique is proposed and studied for the discontinuous Galerkin method applied to hyperbolic equations. In order to avoid the use of slope limiters, a streamline diffusion-like term is added to control oscillations for arbitrary element orders. Thus, the scheme combines ideas from both the Runge–Kutta discontinuous Galerkin method [J. Scient. Comput. 16 (2001) 173] and the streamline diffusion method [Comput. Methods Appl. Mech. Engrg. 32 (1982)]. To increase the stability range of the method, the diffusion term is treated implicitly. The result is a scheme with higher order in space with the same stability range as the finite volume method. An optimal relation between the time step and the size of the diffusion coefficient is analyzed for numerical precision. The scheme is implemented using the object oriented programming philosophy based on the environment described in [Comput. Methods Appl. Mech. Engrg. 150 (1997)]. Accuracy and shock capturing abilities of the method are analyzed in terms of two bidimensional model problems: the rotating cone and the backward facing step problem for the Euler equations of gas dynamics.

Quadrature-Free Implementation of Discontinuous Galerkin Method for Hyperbolic Equations

A discontinuous Galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions. This approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent in the discontinuous Galerkin method. Test problems include the linear and nonlinear one-dimensional scalar advection of smooth initial value problems that are discretized by using unstructured grids with varying degrees of smoothness and regularity, and two-dimensional linear Euler solutions on unstructured grids.

Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations

A discontinuous Galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions. This approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent to the discontinuous Galerkin method. Test problems include both linear and nonlinear one-dimensional scalar advection of both smooth and discontinuous initial value problems, two-dimensional scalar advection of smooth initial value problems discretized by using unstructured grids with varying degrees of smoothness and regularity, and two-dimensional linear Euler solutions on unstructured grids.

An optimal-order error estimate for the discontinuous Galerkin method

In this paper a new approach is developed for analyzing the discontinuous Galerkin method for hyperbolic equations. For a model problem in R 2 {R^2} , the method is shown to converge at a rate O ( h n + 1 ) O({h^{n + 1}}) when applied with nth degree polynomial approximations over a semiuniform triangulation, assuming sufficient regularity in the solution.

Estimates away from a discontinuity for dissipative Galerkin methods for hyperbolic equations

We consider the approximate solution of the initial value problem \[ ∂ u ∂ t = ∂ u ∂ x , u ( x , 0 ) = v ( x ) , \frac {{\partial u}}{{\partial t}} = \frac {{\partial u}}{{\partial x}},\quad u(x,0) = v(x), \] by a dissipative Galerkin method. When v is taken to have a jump discontinuity at zero, that discontinuity will propagate along x + t = 0 x + t = 0 , in the true solution u. Estimates in L 2 {L_2} and L ∞ {L_\infty } of the pollution effects of the discontinuity are found. These estimates show those effects to decay exponentially in h − 1 {h^{ - 1}} in regions a fixed distance d from the discontinuity and exponentially in d for fixed h.

A Simplified Galerkin Method for Hyperbolic Equations

A Simplified Galerkin Method for Hyperbolic Equations

A simplified Galerkin method for hyperbolic equations

We modify a Galerkin method for nonlinear hyperbolic equations so that it becomes a simpler method of lines, which may be viewed as a collocation method. The high order of accuracy is preserved. We present a linear wave analysis of the scheme and discuss some aspects of nonlinear problems. Our numerical experiments indicate that the addition of a proper artificial viscosity makes the method competitive and the common difference schemes, even when the solution has discontinuities.