The brassy timbre associated with loudly played trombone notes is due to the accumulated effects of nonlinear wave propagation. Hirschberg et al. experimentally showed in [J. Acoust. Soc. Am. 99, 1754- 1758] that wave steepening of such tones can be strong enough to produce shock waves. In this paper, pressure measurements collected along the trombone bore are presented and then numerically simu-lated using a three-dimensional time-domain model. The numerical solutions reveal that shock waves do form within the computational instrument within the vicinity of the approximated shock distance. Crown Copyright (c) 2023 Published by Elsevier Ltd. All rights reserved.

Limiters that are used to stabilise the discontinuous Galerkin method fail to preserve its angular momentum conservation property. We propose a simple technique that restores the conservation of angular momentum while preserving the second-order accuracy of the scheme when paired with an explicit time integrator. Numerical experiments confirm the conservation property of the technique paired with two limiters [Giuliani and Krivodonova. 2018. "Analysis of Slope Limiters on Unstructured Triangular Meshes." Journal of Computational Physics 374: 1-26; Giuliani and Krivodonova. 2019. "A Moment Limiter for the Discontinuous Galerkin Method on Unstructured Triangular Meshes." SIAM Journal on Scientific Computing 41 (1): A508-A537]. We show that the technique eliminates numerical artifacts that arise from limiting solutions to the Euler equations.

The length and bore geometry of musical instruments directly influences the quality of sound that can be produced. In brass instruments, nonlinear effects from finite-amplitude wave propagation can lead to wave distortion giving sounds a brassy timbre [3, 5, 14, 20, 26]. In this paper, we propose a three-dimensional model to describe nonlinear wave propagation in a trumpet and investigate the importance of the mouthpiece shank geometry. Time pressure waveforms corresponding to B_3(b) and B_4(b) notes were recorded at the mouthpiece shank and used as inputs for our model. To describe the motion of compressible inviscid fluid, we numerically solved the compressible Euler equations using the discontinuous Galerkin method. To validate our approach, the numerical results were compared to the recorded musical notes outside the bell of the trumpet. Simulations were performed on computational trumpets where different bore geometries were considered. Our results demonstrate that the shape of the narrow region near mouthpiece greatly influences the wave propagation and accuracy of the trumpet model.

We propose a moment limiter of arbitrary high order for the discontinuous Galerkin method on unstructured triangular meshes. The limiter works by hierarchically limiting solution coefficients (moments) by comparing them to reconstructed directional derivatives along specific directions. Limiting along these directions is performed using one-dimensional minmod slope limiters. The stencil used to reconstruct directional derivatives consists of only eight mesh elements and can be computed in the pre-processing stage. Due to a low number of operations involved in limiting, the limiter takes only a fraction of the total computing time for the modal discontinuous Galerkin method. We present numerical examples showing that limited solutions retain the theoretical rate of convergence and are robust in the presence of discontinuities. Crown Copyright (C) 2021 Published by Elsevier Inc. All rights reserved.

The total variation diminishing (TVD) property is an important tool for ensuring nonlinear stability and convergence of numerical solutions of one-dimensional scalar conservation laws. However, it proved to be challenging to extend this approach to two-dimensional problems. Using the anisotropic definition for discrete total variation (TV), it was shown in \cite{Goodman} that TVD solutions of two-dimensional hyperbolic equations are at most first order accurate. We propose to use an alternative definition resulting from a full discretization of the semi-discrete Raviart-Thomas TV. We demonstrate numerically using the second order discontinuous Galerkin method that limited solutions of two-dimensional hyperbolic equations are TVD in means when total variation is computed using the new definition.

Time pressure waveforms associated with four musical notes produced at different volumes on a trombone were measured and then reproduced using a computational model. Special care was taken to accurately model the change in the trombone’s cross-sectional area. An axisymmetric formulation of the compressible Euler equations was used and then numerically solved using the discontinuous Galerkin method. To evaluate the chosen model, the numerical solutions were compared against the measured data collected outside the bell. We found that accounting for the nonlinear behaviour for both high and low sound volumes yielded a good quantitative match between the computed and measured tones. For all four notes, once the sound pressure level drops 30 dB below the main peak, the computed pressure overestimated the measured spectral components.

Mathematical models of a complex physicochemical underground coal gasification process involve multidimensional non-linear partial differential equations (PDEs). Therefore, researchers always seek a suitable numerical scheme that would give a good balance between accuracy of the solution and computational complexity. In this paper, the non-linear PDEs describing energy and mass balances of both coal and char are solved by Galerkin finite element method (GFEM). The infinite-dimensional spatial domain is transformed into a finite number of elements, whose dynamics are governed by a system of ordinary differential equations (ODEs). Owing to the execution of GFEM, the time domain and space-dependent first-order ODEs for solids (coal and char) and gases are solved numerically to find a solution of the UCG process. The resultant syngas compositions and the calibrated heating value are influenced by the operation parameters and the type of oxidant used. The results are compared with the experimental data obtained from the Thar coal UCG site, and with existing work, based on the finite difference method (FDM) for the one dimensional (1D) model. The simulated results along with the quantitative analysis show the superiority of the GFEM model over the FDM model.

When the discontinuous Galerkin (DG) method is applied to hyperbolic problems in two dimensions on triangular meshes and paired with an explicit time integration scheme, an exact CFL condition is not known. The stability condition which is most usually implemented involves scaling the time step by the smallest radius of the inscribed circle in every cell. However, this is known to not provide a tight bound on the largest possible stable time step in some cases. In this paper, we apply the DG method to a simple linear problem and derive a PDE which is satisfied by the numerical solution itself. By applying classical Fourier analysis to the solutions of this PDE we find a natural scaling of the spectrum of the DG spatial operator by a parameter hj, which can be seen to be the width of the cell Ωj along the characteristic direction of flow. We use this parameter to propose a new CFL condition and show through several numerical examples that we are able to select significantly larger time steps than usually obtained using the inscribed radii of the computational cells.

We present novel algorithms for cell-based adaptive mesh refinement on unstructured meshes of triangles on graphics processing units. Our implementation makes use of improved memory management techniques and a coloring algorithm for avoiding race conditions. Both the solver and AMR algorithms are entirely implemented on the GPU, with negligible communication between device and host. We show that the overhead of the AMR subroutines is small compared to the high order solver and that the proportion of total runtime spent adaptively refining the mesh decreases with the order of approximation. We apply our code to a number of benchmark problems as well as more recently proposed problems for the Euler equations that require extremely high resolution. We present the solution to a shock reflection problem that addresses the von Neumann triple point paradox with an accurately computed triple point location. Finally, we present the first solution on the full Euler equations to the problem of shock disappearance and self-similar diffraction of weak shocks around thin films.

We propose a second order limiter for the discontinuous Galerkin method applied to hyperbolic conservation laws. We prove that under a suitable time step restriction, the limiter enforces the local maximum principle for linear and nonlinear scalar equations. The limiter works by finding directions in which the solution coefficients can be separated and limits them independently of one another by comparing to forward and backward reconstructed differences. The limiter has a precomputed stencil of constant size, which provides computational advantages in terms of implementation and run time. We provide examples that demonstrate stability and second order accuracy of solutions.

An axisymmetric model to simulate the evolution of nonlinear waves through a trumpet is presented. In particular, we simulate the time pressure waveform of a musical note as it travels through the instrument. The flare expansion and curvature of the initial tubing near the mouthpiece shank is carefully modeled. For the mathematical model, we chose the compressible Euler equations and solved them numerically using a GPU implementation of the discontinuous Galerkin method. We compare our numerical results with a full three-dimensional model. We find that axisymmetric simulations exhibit less numerical diffusion while providing better resolution without additional mesh refinement. Moreover, axisymmetric simulations significantly reduce runtime and required memory.

We show that the theory for strong stability preserving (SSP) time stepping methods employed with the method of lines-type discretizations of hyperbolic conservation laws may result in overly stringent time step restrictions. We analyze a fully discrete finite volume method with slope reconstruction and a second order SSP Runge–Kutta time integrator to show that the maximum stable time step can be increased over the SSP limit. Numerical examples indicate that this result extends to two-dimensional problems on triangular meshes.

We analyze the stability and accuracy of second order limiters for the discontinuous Galerkin method on unstructured triangular meshes. We derive conditions for a limiter such that the numerical solution preserves second order accuracy and satisfies the local maximum principle. This leads to a new measure of cell size that is approximately twice as large as the radius of the inscribed circle. It is shown with numerical experiments that the resulting bound on the time step is tight. Finally, we consider various combinations of limiting points and limiting neighborhoods and present numerical experiments comparing the accuracy, stability, and efficiency of the corresponding limiters.

We present a novel approach for the indirect generation of unstructured quadrilateral meshes. The algorithm first creates a mesh of unstructured triangles, then partitions the edges of the triangular mesh into three groups with a fast coloring algorithm. An all-quadrilateral mesh is obtained on the domain interior by removing all edges of a particular color. The resulting unstructured quadrilateral mesh is then post-processed using topological optimization techniques for quadrilateral meshes. We apply our algorithm to a number of meshes and demonstrate that it is consistently one to two orders of magnitude faster than a widely used recombination algorithm.

We apply the discontinuous Galerkin finite element method with a degree p polynomial basis to the linear advection equation and derive a PDE which the numerical solution solves exactly. We use a Fourier approach to derive polynomial solutions to this PDE and show that the polynomials are closely related to the p/p+1 Padé approximant of the exponential function. We show that for a uniform mesh of N elements there exist (p+1)N independent polynomial solutions, N of which can be viewed as physical and pN as non-physical. We show that the accumulation error of the physical mode is of order 2p+1 . In contrast, the non-physical modes are damped out exponentially quickly. We use these results to present a simple proof of the superconvergence of the DG method on uniform grids as well as show a connection between spatial superconvergence and the superaccuracies in dissipation and dispersion errors of the scheme. Finally, we show that for a class of initial projections on a uniform mesh, the superconvergent points of the numerical error tend exponentially quickly towards the downwind based Radau points.

We propose an algorithm for preventing race conditions in the evaluation of the surface integral contributions in edge-based CFD solvers by coloring the faces (or edges) of the computational mesh. We use a partitioning algorithm that separates the edges of triangular elements into three groups, the faces of quadrilateral and tetrahedral elements into four groups, and the faces of hexahedral elements into six groups. Our method is also applicable to hybrid meshes. We then extend this partitioning to adaptively refined, nonconforming meshes. We use the coloring to reduce code memory requirements by eliminating buffering. The coloring is also used to renumber and reorder elemental data to optimize reading and writing to memory, thus reducing access latencies and accelerating computations. Crown Copyright (C) 2017 Published by Elsevier B.V. All rights reserved.

We propose a way of preventing race conditions in the evaluation of the surface integral contribution in discontinuous Galerkin and finite volume flow solvers by coloring the edges (or faces) of the computational mesh. In this work we use a partitioning algorithm that separates the edges of triangular elements into three groups and the faces of quadrangular and tetrahedral elements into four groups; we then extend this partitioning to adaptively refined, nonconforming meshes. We use the ascribed coloring to reduce code memory requirements and optimize accessing the elemental data in memory. This process reduces memory access latencies and speeds up computations on graphics processing units.

Simplifying a three-dimensional problem of simulating sound propagation in musical instruments is frequently done by exploiting axial symmetry and reducing the problem to one or two dimensions. We examine if such dimension reduction is valid. We numerically solve the equations of motion of compressible gases using the discontinuous Galerkin method to model nonlinear sound propagation inside a trumpet. The numerical results in two and three dimensions are then compared with experimental data. Experiments were carried out on a trumpet in which the sound pressure waves of the B3 b and B4 b notes played at forte were recorded. We found that it is crucial to consider the problem with all three spatial dimensions to ensure reflections in the bell region are properly modelled. Additionally, the shape of flare must be carefully approximated to compute the propagating waves accurately.

A model for nonlinear sound wave propagation for the trumpet is proposed. Experiments have been carried out to measure the sound pressure waveforms of the [Formula: see text] and [Formula: see text] notes played forte. We use these pressure measurements at the mouthpiece as an input for the proposed model. The compressible Euler equations are used to incorporate nonlinear wave propagation and compressibility effects. The equations of motion are solved using the discontinuous Galerkin method (DGM) and the suitability of this method is assessed. The third spatial dimension is neglected and the consequences for such an assumption are examined. The numerical experiments demonstrate the validity of this approach. We obtain a good match between experimental and numerical data after the dimensionality of the problem is taken into account.

This paper revisits the stochastic near-optimal control problem considered in Zhou (1998), where the stochastic system is given by a controlled stochastic differential equation with the control variable taking values in a general control space and entering both the drift and diffusion coefficients. A necessary condition of near-optimality is derived using Ekeland’s variational principle, spike variation techniques, and some delicate estimates for the state and the adjoint processes. We improve the error bound of order from “almost” ε13 in Zhou (1998) to “exactly” ε13.