Modeling of acoustic wave propagation in time-domain using the discontinuous Galerkin method – A comparison with measurements

Simulations of acoustic wave propagation in time-domain are presented. In the simulations, the discontinuous Galerkin method for spatial derivatives and the low-storage Runge–Kutta approach for time derivatives are used. Three different simulation cases are studied. First, the directivity of loudspeaker is simulated. In the second case, acoustic wave propagation in free space is studied using a short pulse. In the last case, acoustic wave scattering from a metallic cylinder is simulated. All simulation results are compared with measurement results. The measurements for the acoustic wave scattering from the metallic cylinder are made in 2D planes using an automated measurement system. Comparison between the simulation and measurement results are made both temporally and spatially and a good agreement between the simulation and measurement results is found. The results suggest that the discontinuous Galerkin method coupled with the low-storage Runge–Kutta approach is a viable tool for modeling acoustic wave propagation in the time-domain.     

A new time–space domain high-order finite-difference method for the acoustic wave equation

A new unified methodology was proposed in Finkelstein and Kastner (2007) [39] to derive spatial finite-difference (FD) coefficients in the joint time–space domain to reduce numerical dispersion. The key idea of this method is that the dispersion relation is completely satisfied at several designated frequencies. We develop this new time–space domain FD method further for 1D, 2D and 3D acoustic wave modeling using a plane wave theory and the Taylor series expansion. New spatial FD coefficients are frequency independent though they lead to a frequency dependent numerical solution. We prove that the modeling accuracy is 2nd-order when the conventional (2M)$(2M)$th-order space domain FD and the 2nd-order time domain FD stencils are directly used to solve the acoustic wave equation. However, under the same discretization, the new 1D method can reach (2M)$(2M)$th-order accuracy and is always stable. The 2D method can reach (2M)$(2M)$th-order accuracy along eight directions and has better stability. Similarly, the 3D method can reach (2M)$(2M)$th-order accuracy along 48 directions and also has better stability than the conventional FD method. The advantages of the new method are also demonstrated by the results of dispersion analysis and numerical modeling of acoustic wave equation for homogeneous and inhomogeneous acoustic models. In addition, we study the influence of the FD stencil length on numerical modeling for 1D inhomogeneous media, and derive an optimal FD stencil length required to balance the accuracy and efficiency of modeling. A new time–space domain high-order staggered-grid FD method for the 1D acoustic wave equation with variable densities is also developed, which has similar advantages demonstrated by dispersion analysis, stability analysis and modeling experiments. The methodology presented in this paper can be easily extended to solve similar partial difference equations arising in other fields of science and engineering.     

An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection–diffusion equations

In this paper, we present hybridizable discontinuous Galerkin methods for the numerical solution of steady and time-dependent nonlinear convection–diffusion equations. The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary. Applying the Newton–Raphson procedure and the hybridization technique, we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every Newton iteration. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. We then extend the method to time-dependent problems by approximating the time derivative by means of backward difference formulae. When the time-marching method is (p+1)$(p+1)$th order accurate and when polynomials of degree p?0$p?0$ are used to represent the scalar variable, each component of the flux and the approximate trace, we observe that the approximations for the scalar variable and the flux converge with the optimal order of p+1$p+1$ in the L²$L^2$-norm. Finally, we apply element-by-element postprocessing schemes to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous interelement normal component, is shown to converge with order p+1$p+1$ in the L²$L^2$-norm. The new approximate scalar variable is shown to converge with order p+2$p+2$ in the L²$L^2$-norm. The postprocessing is performed at the element level and is thus much less expensive than the solution procedure. For the time-dependent case, the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required. Extensive numerical results are provided to demonstrate the performance of the present method.     

A discontinuous Galerkin method for the Vlasov–Poisson system

A discontinuous Galerkin method for approximating the Vlasov–Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates of the approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov–Poisson system.     

An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations

We present an implicit high-order hybridizable discontinuous Galerkin method for the steady-state and time-dependent incompressible Navier–Stokes equations. The method is devised by using the discontinuous Galerkin discretization for a velocity gradient-pressure–velocity formulation of the incompressible Navier–Stokes equations with a special choice of the numerical traces. The method possesses several unique features which distinguish itself from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Moreover, if the augmented Lagrangian method is used to solve the linearized system, the globally coupled unknowns become the approximate trace of the velocity only. Second, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of k + 1 in the L²-norm, when polynomials of degree k?0 are used for all components of the approximate solution. And third, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, H (div)-conforming, and converges with order k + 2 for k ? 1 and with order 1 for k = 0 in the L²-norm. Moreover, a novel and systematic way is proposed for imposing boundary conditions for the stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the method. This can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method. Extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.     

A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations

The Vlasov–Poisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that self-interacts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh. In particular, we develop in this work a method that discretizes the 1 + 1 Vlasov–Poisson system via a high-order discontinuous Galerkin (DG) method in phase space, and an operator split, semi-Lagrangian method in time. Second-order accuracy in time is relatively easy to achieve via Strang operator splitting. With additional work, using higher-order splitting and a higher-order method of characteristics, we also demonstrate how to push this scheme to fourth-order accuracy in time. We show how to resolve all of the Lagrangian dynamics in such a way that mass is exactly conserved, positivity is maintained, and high-order accuracy is achieved. The Poisson equation is solved to high-order via the smallest stencil local discontinuous Galerkin (LDG) approach. We test the proposed scheme on several standard test cases.     

Optimal Runge–Kutta schemes for discontinuous Galerkin space discretizations applied to wave propagation problems

We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge–Kutta time integrators, with the aim of deriving optimal Runge–Kutta schemes for wave propagation applications. We review relevant Runge–Kutta methods from literature, and consider schemes of order q from 3 to 4, and number of stages up to q + 4, for optimization. From a user point of view, the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method; two scenarios are defined. In the first one, the element size is totally free, and a 8-stage, fourth-order Runge–Kutta scheme is found to minimize a cost measure depending on both accuracy and stability. In the second one, the elements are assumed to be constrained to such a small size by geometrical features of the computational domain, that accuracy is disregarded. We then derive one 7-stage, third-order scheme and one 8-stage, fourth-order scheme that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed, and the benefits are illustrated with two examples. For each of these Runge–Kutta methods, we provide the coefficients for a 2N-storage implementation, along with the information needed by the user to employ them optimally.     

A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations

In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton–Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case with constant coefficients, the method is equivalent to the discontinuous Galerkin method for conservation laws. Thus, stability and error analysis are obtained under the framework of conservation laws. For both convex and noneconvex Hamiltonian, optimal (k + 1)th order of accuracy for smooth solutions are obtained with piecewise kth order polynomial approximations. The scheme is numerically tested on a variety of one and two dimensional problems. The method works well to capture sharp corners (discontinuous derivatives) and have the solution converges to the viscosity solution.     

Space–time discontinuous Galerkin finite element method for two-fluid flows

A novel numerical method for two-fluid flow computations is presented, which combines the space–time discontinuous Galerkin finite element discretization with the level set method and cut-cell based interface tracking. The space–time discontinuous Galerkin (STDG) finite element method offers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local hp-refinement. The front tracking is incorporated via cut-cell mesh refinement to ensure a sharp interface between the fluids. To compute the interface dynamics the level set method (LSM) is used because of its ability to deal with merging and breakup. Also, the LSM is easy to extend to higher dimensions. Small cells arising from the cut-cell refinement are merged to improve the stability and performance. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative. The numerical method is applied to one and two dimensional two-fluid test problems using the Euler equations.     

Nonuniform time-step Runge–Kutta discontinuous Galerkin method for Computational Aeroacoustics

With many superior features, Runge–Kutta discontinuous Galerkin method (RKDG), which adopts Discontinuous Galerkin method (DG) for space discretization and Runge–Kutta method (RK) for time integration, has been an attractive alternative to the finite difference based high-order Computational Aeroacoustics (CAA) approaches. However, when it comes to complex physical problems, especially the ones involving irregular geometries, the time step size of an explicit RK scheme is limited by the smallest grid size in the computational domain, demanding a high computational cost for obtaining time accurate numerical solutions in CAA. For computational efficiency, high-order RK method with nonuniform time step sizes on nonuniform meshes is developed in this paper. In order to ensure correct communication of solutions on the interfaces of grids with different time step sizes, the values at intermediate-stages of the Runge–Kutta time integration on the elements neighboring such interfaces are coupled with minimal dissipation and dispersion errors. Based upon the general form of an explicit p-stage RK scheme, a linear coupling procedure is proposed, with details on the coefficient matrices and execution steps at common time-levels and intermediate time-levels. Applications of the coupling procedures to Runge–Kutta schemes frequently used in simulation of fluid flow and acoustics are given, including the third-order TVD scheme, and low-storage low dissipation and low dispersion (LDDRK) schemes. In addition, an analysis on the stability of coupling procedures on a nonuniform grid is carried out. For validation, numerical experiments on one-dimensional and two-dimensional problems are presented to illustrate the stability and accuracy of proposed nonuniform time-step RKDG scheme, as well as the computational benefits it brings. Application to a one-dimensional nonlinear problem is also investigated.     

An efficient high-order boundary element method for nonlinear wave–wave and wave-body interactions

A highly efficient high-order boundary element method is developed for the numerical simulation of nonlinear wave–wave and wave-body interactions in the context of potential flow. The method is based on the framework of the quadratic boundary element method (QBEM) for the boundary integral equation and uses the pre-corrected fast Fourier transform (PFFT) algorithm to accelerate the evaluation of far-field influences of source and/or normal dipole distributions on boundary elements. The resulting PFFT–QBEM reduces the computational effort of solving the associated boundary-value problem from O(N^2～3) (with the traditional QBEM) to O(N ln N) where N represents the total number of boundary unknowns. Significantly, it allows for reliable computations of nonlinear hydrodynamics useful in ship design and marine applications, which are forbidden with the traditional methods on the presently available computing platforms. The formulation and numerical issues in the development and implementation of the PFFT–QBEM are described in detail. The characteristics of accuracy and efficiency of the PFFT–QBEM for various boundary-value problems are studied and compared to those of the existing accelerated (lower- and higher-order) boundary element methods. To illustrate the usefulness of the PFFT–QBEM, it is applied to solve the initial boundary-value problem in the generation of three-dimensional nonlinear waves by a moving ship hull. The predicted wave profile and resistance on the ship are compared to available experimental measurements with satisfactory agreements.     

A reconstructed discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids

A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier–Stokes equations on arbitrary grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier–Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on arbitrary grids. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG method is able to deliver the same accuracy as the well-known Bassi–Rebay II scheme, at a half of its computing costs for the discretization of the viscous fluxes in the Navier–Stokes equations, clearly demonstrating its superior performance over the existing DG methods for solving the compressible Navier–Stokes equations.     

Uncertainty propagation in SEA for structural–acoustic coupled systems with non-deterministic parameters

Considering limited available information on uncertainties in structural - acoustic coupled systems, two methods namely the vertex method and the Legendre orthogonal polynomial based method for predicting their dynamic behavior are developed based on the Statistical Energy Analysis (SEA) approach. For the vertex method, an efficient program for determining coordinates of all vertices of the rectangular spanned by entries of the involved interval input vector is coded, which is well suited for an interval input vector in arbitrary dimension. Instead of calculating the extremum of the response of interest, a method for determining its minimal and maximal point vectors dimension by dimension with respect to uncertain parameters is proposed based on the Legendre orthogonal polynomial approximation. Following the theoretical analysis of the accuracy and efficiency of the proposed methods, their validation is performed by one numerical example and two applications.     

A hybrid discontinuous Galerkin method for computing the ground state solution of Bose–Einstein condensates

A numerical method for computing the ground state solution of Bose–Einstein condensates modeled by the Gross–Pitaevskii equation is presented. In this method, the three-dimensional computational domain is divided into hexahedral elements in which the solution is approximated by a sum of basis functions. Both polynomial and plane wave bases are considered for this purpose, and Lagrange multipliers are introduced to weakly enforce the interelement continuity of the solution. The ground state is computed by an iterative procedure for minimizing the energy. The performance results obtained for several numerical experiments demonstrate that the proposed method is more computationally efficient than similar solution approaches based on the standard higher-order finite element method.     

A coupled solid–fluid method for modelling subduction

We present a novel dynamic approach for solid–fluid coupling by joining two different numerical methods: the boundary-element method (BEM) and the finite element method (FEM). The FEM results describe the thermomechanical evolution of the solid while the fluid is solved with the BEM. The bidirectional feedback between the two domains evolves along a Lagrangian interface where the FEM domain is embedded inside the BEM domain. The feedback between the two codes is based on the calculation of a specific drag tensor for each boundary on finite element. The approach is presented here to solve the complex problem of the descent of a cold subducting oceanic plate into a hot fluid-like mantle. The coupling technique is shown to maintain the proper energy dissipation caused by the important secondary induced mantle flow induced by the lateral migrating of the subducting plate. We show how the method can be successfully applied for modelling the feedback between deformation of the oceanic plate and the induced mantle flow. We find that the mantle flow drag is singular at the edge of the retreating plate causing a distinct hook shape. In nature, such hooks can be observed at the northern end of the Tonga trench and at the southern perimeter, of the South American trench.     

Direct discontinuous Galerkin method for the generalized Burgers—Fisher equation

In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge-Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method.     

A class of discontinuous Petrov–Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L²-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed.     

An interface treating technique for compressible multi-medium flow with Runge–Kutta discontinuous Galerkin method

The high-order accurate Runge–Kutta discontinuous Galerkin (RKDG) method is applied to the simulation of compressible multi-medium flow, generalizing the interface treating method given in Chertock et al. (2008) [9]. In mixed cells, where the interface is located, Riemann problems are solved to define the states on both sides of the interface. The input states to the Riemann problem are obtained by extrapolation to the cell boundary from solution polynomials in the neighbors of the mixed cell. The level set equation is solved by using a high-order accurate RKDG method for Hamilton–Jacobi equations, resulting in a unified DG solver for the coupled problem. The method is conservative if we include the states in the mixed cells, which are however not used in the updating of the numerical solution in other cells. The states in the mixed cells are plotted to better evaluate the conservation errors, manifested by overshoots/undershoots when compared with states in neighboring cells. These overshoots/undershoots in mixed cells are problem dependent and change with time. Numerical examples show that the results of our scheme compare well with other methods for one and two-dimensional problems. In particular, the algorithm can capture well complex flow features of the one-dimensional shock entropy wave interaction problem and two-dimensional shock–bubble interaction problem.