Praveen Chandrashekar

Astrophysical hydrodynamics with a high-order discontinuous Galerkin scheme and adaptive mesh refinement

Solving the Euler equations of ideal hydrodynamics as accurately and efficiently as possible is a key requirement in many astrophysical simulations. It is therefore important to continuously advance the numerical methods implemented in current astrophysical codes, especially also in light of evolving computer technology, which favours certain computational approaches over others. Here we introduce the new adaptive mesh refinement (AMR) code TENET, which employs a high order discontinuous Galerkin (DG) scheme for hydrodynamics. The Euler equations in this method are solved in a weak formulation with a polynomial basis by means of explicit Runge-Kutta time integration and Gauss-Legendre quadrature. This approach offers significant advantages over commonly employed second order finite volume (FV) solvers. In particular, the higher order capability renders it computationally more efficient, in the sense that the same precision can be obtained at significantly less computational cost. Also, the DG scheme inherently conserves angular momentum in regions where no limiting takes place, and it typically produces much smaller numerical diffusion and advection errors than a FV approach. A further advantage lies in a more natural handling of AMR refinement boundaries, where a fall-back to first order can be avoided. Finally, DG requires no wide stencils at high order, and offers an improved data locality and a focus on local computations, which is favourable for current and upcoming highly parallel supercomputers. We describe the formulation and implementation details of our new code, and demonstrate its performance and accuracy with a set of two- and three-dimensional test problems. The results confirm that DG schemes have a high potential for astrophysical applications.

Entropy Stable Finite Volume Scheme for Ideal Compressible MHD on 2-D Cartesian Meshes

We present a finite volume scheme for ideal compressible magnetohydrodynamic (MHD) equations on two-dimensional Cartesian meshes. The semidiscrete scheme is constructed to be entropy stable by using the symmetrized version of the equations as introduced by Godunov. We first construct an entropy conservative scheme for which sufficient condition is given and we also derive a numerical flux satisfying this condition. Second, following a standard procedure, we make the scheme entropy stable by adding dissipative flux terms using jumps in entropy variables. A semi-discrete high resolution scheme is constructed that preserves the entropy stability of the first order scheme. We demonstrate the robustness of this new scheme on several standard MHD test cases.

Error Estimation for High Speed Flows Using Continuous and Discrete Adjoints

In this work, the utility of the adjoint equations in error estimation of functional outputs and goal-oriented mesh adaptation is investigated with specific emphasis towards application to high speed flows with strong shocks. Continuous and discrete adjoint formulations are developed for the compressible Euler equations and the accuracy and robustness of the implementation is assessed by evaluating adjoint-computed sensitivities. The two-grid approach of Venditti and Darmofal 1 - where the flow and adjoint solutions on a baseline mesh are processed to estimate the functional on a finer mesh - is used for error estimation and mesh adaptation. Using a carefully designed set of test cases for the quasi one dimensional Euler equations, it is shown that discrete adjoints can be used to estimate the fine grid functional more accurately, whereas continuous adjoints are marginally better at estimating the analytical value of the functional when the flow and adjoint solutions are well resolved. These observations appear to be true in multi-dimensional inviscid flows, but the distinction is not as clear. The discrete adjoint is shown to be robust when applied to goal oriented mesh adaptation in a flow with multiple shocks, and hence offers promise as a viable strategy to control the numerical error in Hypersonic flow applications.

Well-Balanced Nodal Discontinuous Galerkin Method for Euler Equations with Gravity

We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss–Lobatto–Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh composed of quadrilateral cells and for any gravitational potential. It is applied on several examples to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.

An Efficient Numerical Algorithm for the Inversion of an Integral Transform Arising in Ultrasound Imaging

We present an efficient and novel numerical algorithm for inversion of transforms arising in imaging modalities such as ultrasound imaging, thermoacoustic and photoacoustic tomography, intravascular imaging, non-destructive testing, and radar imaging with circular acquisition geometry. Our algorithm is based on recently discovered explicit inversion formulas for circular and elliptical Radon transforms with radially partial data derived by Ambartsoumian, Gouia-Zarrad, Lewis and by Ambartsoumian and Krishnan. These inversion formulas hold when the support of the function lies on the inside (relevant in ultrasound imaging, thermoacoustic and photoacoustic tomography, non-destructive testing), outside (relevant in intravascular imaging), both inside and outside (relevant in radar imaging) of the acquisition circle. Given the importance of such inversion formulas in several new and emerging imaging modalities, an efficient numerical inversion algorithm is of tremendous topical interest. The novelty of our non-iterative numerical inversion approach is that the entire scheme can be pre-processed and used repeatedly in image reconstruction, leading to a very fast algorithm. Several numerical simulations are presented showing the robustness of our algorithm.

Vertex-centroid finite volume scheme on tetrahedral grids for conservation laws

Vertex-centroid schemes are cell-centered finite volume schemes for conservation laws which make use of both centroid and vertex values to construct high-resolution schemes. The vertex values must be obtained through a consistent averaging (interpolation) procedure while the centroid values are updated by the finite volume scheme. A modified interpolation scheme is proposed which is better than existing schemes in giving positive weights in the interpolation formula. A simplified reconstruction scheme is also proposed which is also more efficient and leads to more robust schemes for discontinuous problems. For scalar conservation laws, we develop limited versions of the schemes which are stable in maximum norm by constructing suitable limiters. The schemes are applied to compressible flows governed by the Euler equations of inviscid gas dynamics.

Positivity-preserving high-order discontinuous Galerkin schemes for Ten-Moment Gaussian closure equations

Euler equations for compressible flows treats pressure as a scalar quantity. However, for several applications this description of pressure is not suitable. Many extended model based on the higher moments of Boltzmann equations are considered to overcome this issue. One such model is Ten-Moment Gaussian closure equations, which treats pressure as symmetric tensor. In this work, we develop a higher-order, positivity preserving Discontinuous Galerkin (DG) scheme for Ten-Moment Gaussian closure equations. The key challenge is to preserve positivity of density and symmetric pressure tensor. This is achieved by constructing a positivity limiter. In addition to preserve positivity, the scheme also ensures the accuracy of the approximation for smooth solutions. The theoretical results are then verified using several numerical experiments.

Simulating Turbulence Using the Astrophysical Discontinuous Galerkin Code TENET

In astrophysics, the two main methods traditionally in use for solving the Euler equations of ideal fluid dynamics are smoothed particle hydrodynamics and finite volume discretization on a stationary mesh. However, the goal to efficiently make use of future exascale machines with their ever higher degree of parallel concurrency motivates the search for more efficient and more accurate techniques for computing hydrodynamics. Discontinuous Galerkin (DG) methods represent a promising class of methods in this regard, as they can be straightforwardly extended to arbitrarily high order while requiring only small stencils. Especially for applications involving comparatively smooth problems, higher-order approaches promise significant gains in computational speed for reaching a desired target accuracy. Here, we introduce our new astrophysical DG code TENET designed for applications in cosmology, and discuss our first results for 3D simulations of subsonic turbulence. We show that our new DG implementation provides accurate results for subsonic turbulence, at considerably reduced computational cost compared with traditional finite volume methods. In particular, we find that DG needs about 1.8 times fewer degrees of freedom to achieve the same accuracy and at the same time is more than 1.5 times faster, confirming its substantial promise for astrophysical applications.

Goal Oriented Uncertainty Propagation using Stochastic Adjoints

We propose a framework based on the use of adjoint equations to formulate an adaptive sampling strategy for uncertainty quantification for problems governed by algebraic or differential equations involving random parameters. The approach is non-intrusive and makes use of discrete sampling based on collocation on simplex elements in stochastic space. Adjoint or dual equations are introduced to estimate errors resulting from the inexact reconstruction of the solution within the simplex elements. The approach is demonstrated to be accurate in estimating errors in statistical moments of interest and shown to exhibit super-convergence, in accordance with the underlying theoretical rates. Goal-oriented error indicators are then built using the adjoint solution and exploited to identify regions for adaptive sampling. The error-estimation and adaptive refinement strategy is applied to a range of problems including those governed by algebraic equations as well as scalar and systems of ordinary and partial differential equations. The strategy holds promise as a reliable method to set and achieve error tolerances for efficient aleatory uncertainty quantification in complex problems. Furthermore, the procedure can be combined with numerical error estimates in physical space so as to effectively manage a computational budget to achieve the best possible overall accuracy in the results.