# Invited Talks

## Sivaram Ambikasaran: Fast O(N) direct sparse solver for finite difference matrices

We discuss a fast and memory efficient solver for sparse matrices arising from finite difference discretization of elliptic partial differential equations (PDEs). Multifrontal methods based on nested dissection for solving these systems scale as O(N^{3/2}) in 2D and O(N^2) in 3D, which is prohibitively expensive for large scale systems. However, these multifrontal methods can be accelerated to linear complexity by observing that certain dense blocks that arise in the elimination process can be efficiently represented as low-rank matrices, which in-turn can be leveraged to obtain O(N) complexity. The talk will discuss the overall approach and present a couple of examples illustrating the linear scaling of the algorithm in 2D and 3D.

## K. R. Arun: Asymptotic preserving schemes for the shallow water system in the low Froude Number limit

We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate the linear waves implicitly in time and the nonlinear advection part explicitly in time. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

## Aravind Balan: Adjoint-based hp-adaptivity on anisotropic meshes for high-order compressible flow simulations

High-order numerical methods such as Discontinuous Galerkin, Spectral Difference, and Flux Reconstruction etc, which use polynomials that are local to each mesh element to represent the solution field, are becoming increasingly popular in solving convection-dominated flows. This is due to their potential in giving accurate results more efficiently than lower order methods such as the classical Finite Volume methods. In most engineering applications, we are more interested in some specific scalar quantities rather than the full flow details. In the case of aerodynamic flow simulations, these quantities can be lift or drag coefficient. To get accurate values for such target functional quantities, adjoint-based error estimators, along with a high-order solver, have been found to be quite useful. They can identify the mesh elements that contribute the most to the error, and adapting these elements should result in a more accurate target functional. To adapt a mesh element, one can either do mesh refinement (h-adaptation) or polynomial space enrichment (p-adaptation) or both (hp-adaptation). Of these, hp-adaptation offers the most efficient way for adaptation, since one can locally choose between mesh refinement or polynomial space enrichment based on what is more efficient in resolving the local solution features.

We present efficient adjoint-based hp-adaptation methodologies on anisotropic meshes for the recently developed high order Hybridized Discontinuous Galerkin scheme for (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. For anisotropic meshes, we extend the refinement strategy based on an interpolation error estimate, due to Dolejsi, by incorporating an adjoint-based error estimate. Using the two error estimates we determine the size and the shape of the triangular mesh elements on the desired mesh to be used for the subsequent adaptation steps. This is done using the concept of mesh-metric duality, where the metric tensors can encode information about mesh elements, which can be passed to a metric-conforming mesh generator to generate the required anisotropic mesh.

## Dinshaw Balsara: Multidimensional, Self-similar, strongly-Interacting, Consistent (MuSIC) Riemann Solvers – Applications to Divergence-Free MHD and ALE Schemes

Large-scale, multidimensional flow simulations are now commonplace and there is a considerable interest in very accurate algorithms for such simulations. Introducing true multidimensionality in such algorithms is very valuable for a complete representation of the physics of the problem. Reconstruction strategies for hyperbolic PDEs (WENO, DG, PNPM, MOOD) are already fully multidimensional as are methods for their temporal update (RK, ADER). The majority of Riemann solvers are still one-dimensional. The present talk describes the design of multidimensional Riemann solvers and their applicability to higher order schemes.

Such multidimensional Riemann solvers act at the vertices of the mesh, where the multidimensional flow structure becomes visible to the Riemann solver. Instead of two input states, the input states consist of states from all the zones that meet at that vertex. At any zone interface that separates two states, a one dimensional Riemann problem emanates, as always. However, at any vertex, all the adjacent one-dimensional Riemann problems interact to form a strongly interacting state. The strongly interacting state evolves self-similarly in spacetime. By evolving the structure of the strongly interacting state in a set of self-similar variables we show that the structure of the strongly interacting state can be elucidated. Self-similarity is crucially important in the development of multidimensional Riemann solvers (Balsara (2010, 2012, 2014, 2015), Balsara, Dumbser & Abgrall (2014), Balsara & Dumbser (2015), Balsara et al. (2015)). This has prompted the name of MuSIC Riemann solvers, where MuSIC stands for “Multidimensional, Self-similar, strongly-Interacting, Consistent”. For a video introduction to multidimensional Riemann solvers see: http://www.nd.edu/~dbalsara/Numerical-PDE-Course.

Numerical MHD has come into its own in the last several years. MHD forms an involution-constrained system where the magnetic field, once divergence-free, remains so forever. As a result, one has to find a strategy to represent the magnetic field in divergence-free fashion. Keeping the magnetic field divergence-free requires solving the problem on a Yee-type mesh. This necessarily requires identifying the multidimensionally upwinded electric field at the edges of a computational mesh. I proceed to show that recent advances in designing multidimensional Riemann solvers give a unique, multidimensionally-upwinded representation of the electric field.

The benefits of the multidimensional Riemann solver go beyond numerical MHD and apply to any hyperbolic system. I show that the multidimensional Riemann solver gives more isotropic flow on resolution-starved meshes. The permitted CFL number is also increased. Even more importantly, the multidimensional Riemann solver gives us a physically-motivated node solver for any ALE application involving any manner of hyperbolic system. The talk ends with presentation of hydrodynamical and magnetohydrodynamical ALE results at all orders.

## Ankik Kumar Giri: Sectional Methods for Solving Smoluchowski Coagulation Equation

The continuous Smoluchowski coagulation equation (SCE) describes the kinetics of particle growth in which particles can aggregate via binary interaction to form larger particles. This model arises in many fields of science and engineering: kinetic of phase transformations in binary alloys such as segregation of binary alloys, aggregation of red blood cells in biology, fluidized bed granulation processes, aerosol physics, i.e. the evolution of a system of solid or liquid particles suspended in a gas, formation of planets in astrophysics, polymer science and many more.There are several mathematical results on existence and uniqueness of solutions to the SCE for different classes of coagulation kernels. The continuous SCE can be solved analytically only for some specific examples of aggregation or coagulation kernels. In general we need to solve it numerically. Many numerical methods have been proposed to solve the SCE. By implementing most of the numerical methods, we may have a quite satisfactory results for the number density but not for moments. In the recent times, the sectional methods have become computationally very attractive because they not only predict accurately some selected moments of the distribution, but also give satisfactory results for the complete density distribution. Among all sectional methods, the fixed pivot technique is the most extensively used method these days due to its generality and robustness. It predicts the first two moments of the distribution very accurately. Despite, the fact that the first two numerically calculated moments are fairly accurate, the fixed pivot technique consistently over predicts the results of number density as well as its higher moments in the large size range when applied on coarse grids. A step to improve the fixed pivot technique by preserving all advantages of the existing sectional methods has been recently made as the cell average technique.

In this talk, we discuss the convergence analysis of the fixed pivot technique for solving SCE. In particular, we investigate the convergence for five different types of uniform and non-uniform meshes which turns out that the fixed pivot technique is second order convergent on a uniform and non-uniform smooth meshes. Moreover, it yields first order convergence on a locally uniform mesh. Finally, the analysis shows that the method does not converge on an oscillatory and non-uniform random meshes. Mathematical results of the convergence analysis are also demonstrated numerically. Next, we study the cell average technique for solving the same problem by performing a few numerical experiments. Finally, the numerical results obtained are also compared with those of the fixed pivot technique and it is observed that the cell average technique improves the order of convergence on coarse grids.

## Jiten Chandra Kalita: Critical points and vortical structures in the 3D lid driven cavity flow

In this study we endeavor to gain some physical insight into the separated flow in the 3D lid driven cavity by a rigorous topological theory. The computed flow is post processed to identify the critical points in the flow leading to the prediction of separation, reattachment, and vortical structures in the flow. The types of critical points found on the surface flow are in accordance with Poincare's formula.

## Rathish Kumar: A look at the Complete Flux Scheme (CFS) for Advection Diffusion Reaction Equation (ADRE)

CFS is based on the idea of including the source terms in defining the fluxes in the FVM frame work and thereby attempt to come up with improved numerical schemes in handling convection dominated fluid flows. Few results from 1D and 2D case studies related to ADRE and also to SPPDEs will be presented.

## J. C. Mandal: Development of finite volume method for compressible flows

In spite of great advances made over the years, numerical computations of high and very low Mach number flows still remain a challenge today. Computations of high Mach number flows are difficult due to the presence of discontinuities in the solutions. Whereas, computations of very low Mach number flows suffer from low accuracy and convergence problems due to ill-conditioning. The challenges get further compounded in multi-dimensions and bring in uncertainty in the flow computations. In order to ensure robustness and greater reliability in computations, several measures like conservation, up-winding, monotonicity, positivity, entropy conditions and even scaling of disparate wave speeds are introduced into the discretization process. In spite of the care taken in developing numerical methods, many computational problems are still not resolved satisfactorily. Solutions to some of the problems associated with higher order finite volume formulations on unstructured grids, catastrophic failure due to numerical shock instability, genuinely multidimensional up-winding, low and all Mach number flow computations are still eluding the CFD experts. Our personal experience in dealing with some of the above problems will be narrated during the talk.

## Sanyasiraju Yedida: Finite difference type computations using radial basis functions (RBF)

## Natesan Srinivasan: Layer-Adapted Nonuniform Meshes for Singular Perturbation Problems

Singular Perturbation problems (SPPs) arise in various applied fields of science and engineering. The Navier-Stokes equation with a large Reynolds number is one of the most striking examples of SPPs, leading to the idea of boundary layer, due to Prandtl. The solution of SPPs exhibit boundary and interior layers. Classical numerical methods like finite difference/element/volume fail to yield satisfactory numerical results for smaller values of the diffusion parameter on uniform meshes. The maximum point-wise error usually increases as the mesh is refined - because of the presence of the layer - until the mesh diameter is comparable in size to the diffusion parameter. Therefore, it is essential to develop robust numerical schemes for SPPs, which provide uniform convergence numerical results. In this talk, we focus on the construction of layer-adapted nonuniform meshes for these problems. The classical finite difference schemes provide parameter-uniform convergent numerical solutions on these adaptive meshes. Some numerical examples will be provided to validate the theoretical error estimates.

## T. N. Venkatesh: Explorations in use of spectral methods for solution of PDEs

For numerical solution of problems which require a high-order of accuracy and faithful representation across wave numbers, spectral methods are the first choice. Examples are i) the use of Fourier series for Direct Numerical Simulation of turbulence, and ii) Spherical Harmonics for atmospheric General Circulation Models. Interesting examples of simulations using these methods will be presented. In addition, recent work on the formulation of a hybrid Fourier-wavelet transform and its use for solution of PDEs will be discussed.