Discontinuous Galerkin Code Research Articles

Implementation and application of complete radiation boundary conditions: A demonstration

The radiation of energy to the far field is a fundamental feature of many problems in acoustics. Efficient numerical algorithms for simulating sound propagation in the time domain must therefore include convergent near-field domain truncation procedures. For problems modeled by uniform or stratified far fields, complete radiation boundary conditions provide a provably optimal domain truncation algorithm. In particular, they are spectrally convergent in boundary condition order and can be placed quite close to scatterers or other inhomogeneities. However, the mathematical description of the radiation conditions looks complicated in comparison with less-efficient methods, such as perfectly matched layers or simple damping. In this demonstration, we show how complete radiation conditions can be implemented in a standard discontinuous Galerkin code as easily as other schemes. In fact, the implementation is shown to simply involve a single extra element with a nonstandard differentiation matrix.

MUSES: A nonlinear magnetohydrodynamics discontinuous Galerkin code for fusion plasmas

MUSES: A nonlinear magnetohydrodynamics discontinuous Galerkin code for fusion plasmas

Yet Another Tensor Toolbox for Discontinuous Galerkin Methods and Other Applications

The numerical solution of partial differential equations is at the heart of many grand challenges in supercomputing. Solvers based on high-order discontinuous Galerkin (DG) discretisation have been shown to scale on large supercomputers with excellent performance and efficiency if the implementation exploits all levels of parallelism and is tailored to the specific architecture. However, every year new supercomputers emerge and the list of hardware-specific considerations grows simultaneously with the list of desired features in a DG code. Thus, we believe that a sustainable DG code needs an abstraction layer to implement the numerical scheme in a suitable language. We explore the possibility to abstract the numerical scheme as small tensor operations, describe them in a domain-specific language (DSL) resembling the Einstein notation, and to map them to small General Matrix-Matrix Multiplication routines. The compiler for our DSL implements classic optimisations that are used for large tensor contractions, and we present novel optimisation techniques such as equivalent sparsity patterns and optimal index permutations for temporary tensors. Our application examples, which include the earthquake simulation software SeisSol, show that the generated kernels achieve over 50% peak performance of a recent 48-core Skylake system while the DSL considerably simplifies the implementation.

Discontinuous Galerkin solution of the RANS and [formula omitted] equations for natural and bypass transition

Discontinuous Galerkin solution of the RANS and [formula omitted] equations for natural and bypass transition

Interactions of two bubbles along a gaseous interface undergoing the Richtmyer–Meshkov instability in two dimensions

Interactions of two bubbles along a gaseous interface undergoing the Richtmyer–Meshkov instability in two dimensions

Hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity

A considerable amount of attention has been given to discontinuous Galerkin methods for hyperbolic problems in numerical relativity, showing potential advantages of the methods in dealing with hydrodynamical shocks and other discontinuities. This paper investigates discontinuous Galerkin methods for the solution of elliptic problems in numerical relativity. We present a novel hp-adaptive numerical scheme for curvilinear and non-conforming meshes. It uses a multigrid preconditioner with a Chebyshev or Schwarz smoother to create a very scalable discontinuous Galerkin code on generic domains. The code employs compactification to move the outer boundary near spatial infinity. We explore the properties of the code on some test problems, including one mimicking Neutron stars with phase transitions. We also apply it to construct initial data for two or three black holes.

Rapid non-linear finite element analysis of continuous and discontinuous Galerkin methods in MATLAB

MATLAB is adept at the development of concise Finite Element (FE) routines, however it is commonly perceived to be too inefficient for high fidelity analysis. This paper aims to challenge this preconception by presenting two optimised FE codes for both continuous Galerkin (CG) and discontinuous Galerkin (DG) methods. Although this has previously been achieved for linear-elastic problems, no such optimised MATLAB script currently exists, which includes the effects of material non-linearity. To incorporate these elasto-plastic effects, the externally applied load is split into a discrete number of loadsteps. Equilibrium is determined at each loadstep between the externally applied load and the arising internal forces using the Newton–Raphson method. The optimisation of the scripts is primarily achieved using vectorised blocking algorithms, which minimise RAM-to-cache overheads and maximise cache reuse.The optimised codes yielded maximum speed gains of ×25.7 and ×10.1 when compared to the corresponding unoptimised scripts, for CG and DG respectively. It was identified that with increasing refinement of the mesh, the solver time begins to dominate the overall simulation time. This bottleneck has a greater disadvantage on the DG code, predominantly due the asymmetric nature of the global stiffness matrix. The implementation of an efficient solver would see further improvement to the overall run times, particularly for large problems.

External Magnetic Field Effects on Ablation of Current-Driven Foils Using an Extended Magnetohydrodynamics Simulation

We numerically model the ablation process of a 25- $\mu \text{m}$ -thick aluminum foil driven by a pulsed-power machine that provides a 1-MA peak current in a 100-ns zero-to-peak rise time. The extended magnetohydrodynamics simulation is a discontinuous Galerkin code with Cartesian coordinates in 3-D and with 25- $\mu \text{m}$ spatial resolution. We investigate the influence of an external magnetic field normal to the foil surface, $B_{z}$ . During the foil ablation, $B_{z}=1$ T causes more nonuniform distributions of density and current compared to $B_{z}=0$ T. $B_{z}=4$ T delays the generation of surface plasma relative to the 0- and 1-T cases. The understanding of a material’s ablation as it undergoes transition from the solid to plasma phases requires detailed knowledge of a material’s equation of state and conductivity. This paper of warm dense matter and how instabilities propagate from a solid material to plasma motivates improvements to both numerical simulations and experimental diagnostics.

Linear and nonlinear ultrasound simulations using the discontinuous Galerkin method.

A nodal discontinuous Galerkin (DG) code based on the nonlinear wave equation is developed to simulate transient ultrasound propagation. The DG method has high-order accuracy, geometric flexibility, low dispersion error, and excellent scalability, so DG is an ideal choice for solving this problem. A nonlinear acoustic wave equation is written in a first-order flux form and discretized using nodal DG. A dynamic sub-grid scale stabilization method for reducing Gibbs oscillations in acoustic shock waves is then established. Linear and nonlinear numerical results from a two-dimensional axisymmetric DG code are presented and compared to numerical solutions obtained from linear and Khokhlov-Zabolotskaya-Kuznetsov-based simulations in FOCUS. The numerical results indicate that these nodal DG simulations capture nonlinearity, thermoviscous absorption, and diffraction for both flat and focused pistons in homogeneous media.

Nonlinear ultrasound simulations using the discontinuous Galerkin method

Histotripsy with nonlinear ultrasound is an emerging noninvasive therapeutic modality that generates cavitation with high-intensity shock waves to precisely destroy diseased soft tissue. Numerical simulation of these shock waves in nonlinear, absorbing media is needed to characterize histotripsy systems and optimize treatments. We are developing a discontinuous Galerkin code based on the Westervelt equation to simulate transient wave propagation in the brain and skull. The discontinuous Galerkin method is a good choice for this simulation problem since this approach has high-order accuracy, geometric flexibility, low dispersion error, and excellent scalability on massively parallel machines. The Westervelt equation is formulated in a first-order flux form and discretized using a strong form of the discontinuous Galerkin method. Numerical results, both linear and nonlinear, from 1D and 2D discontinuous Galerkin codes, are presented and compared to both analytical and numerical benchmark solutions. In particular, the discontinuous Galerkin method captures nonlinear steepening of a high-intensity pulse with minimal numerical artifacts. The development of a 3D massively parallel code is also briefly discussed. [This work was supported in part by a grant from the Focused Ultrasound Foundation.]

SpECTRE: A task-based discontinuous Galerkin code for relativistic astrophysics

SpECTRE: A task-based discontinuous Galerkin code for relativistic astrophysics

High performance computing aspects of a dimension independent semi-Lagrangian discontinuous Galerkin code

High performance computing aspects of a dimension independent semi-Lagrangian discontinuous Galerkin code

High-order implementation of a non-local transition model in a DG solver for turbomachinery applications

High-order implementation of a non-local transition model in a DG solver for turbomachinery applications

Riemann solutions and spacetime discontinuous Galerkin method for linear elastodynamic contact

Riemann solutions and spacetime discontinuous Galerkin method for linear elastodynamic contact

A parallel, high-order discontinuous Galerkin code for laminar and turbulent flows

A parallel, high-order discontinuous Galerkin code for laminar and turbulent flows

On the Construction and Comparison of Difference Schemes

On the Construction and Comparison of Difference Schemes