Praveen Chandrashekar

Praveen Chandrashekar, Ashish Garg

Vertex-centroid schemes are cell-centered finite volume schemes for conservation laws which make use of vertex values to construct high resolution schemes. The vertex values must be obtained through a consistent averaging (interpolation) procedure. 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 accurate and efficient. 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.

Sujoy Basak, Arpit Babbar, Harish Kumar et al.

The relativistic hydrodynamics (RHD) equations can give rise to solutions which have shocks, contact discontinuities, and other sharp structures, which interact and evolve over time. Capturing these sharp waves effectively requires a mesh with high resolution, making the scheme computationally expensive. In this work, adaptive mesh refinement is used with the high-order Lax-Wendroff flux reconstruction (LWFR) method to solve the system of RHD equations, which is closed with general equations of state. To make the scheme Jacobian-free, the idea of automatic differentiation is incorporated for computing the temporal derivatives in the time average flux approximations. The high-order method is blended with an admissible low-order method at the subcell level to control the Gibbs oscillations and maintain the physical admissibility of the solution. Finally, several test cases involving high Lorentz factors, low densities, low pressures, strong shock waves, and other discontinuities are used to demonstrate the robustness, accuracy, and effectiveness of the proposed method. These simulations are performed with AMR using various linear and curved meshes to show the scheme's efficiency and ability to handle complex geometries.

Shizheng Wen, Arsh Kumbhat, Levi Lingsch et al.

The very challenging task of learning solution operators of PDEs on arbitrary domains accurately and efficiently is of vital importance to engineering and industrial simulations. Despite the existence of many operator learning algorithms to approximate such PDEs, we find that accurate models are not necessarily computationally efficient and vice versa. We address this issue by proposing a geometry aware operator transformer (GAOT) for learning PDEs on arbitrary domains. GAOT combines novel multiscale attentional graph neural operator encoders and decoders, together with geometry embeddings and (vision) transformer processors to accurately map information about the domain and the inputs into a robust approximation of the PDE solution. Multiple innovations in the implementation of GAOT also ensure computational efficiency and scalability. We demonstrate this significant gain in both accuracy and efficiency of GAOT over several baselines on a large number of learning tasks from a diverse set of PDEs, including achieving state of the art performance on three large scale three-dimensional industrial CFD datasets.

Kjetil O. Lye, Siddhartha Mishra, Deep Ray et al.

We present a novel active learning algorithm, termed as iterative surrogate model optimization (ISMO), for robust and efficient numerical approximation of PDE constrained optimization problems. This algorithm is based on deep neural networks and its key feature is the iterative selection of training data through a feedback loop between deep neural networks and any underlying standard optimization algorithm. Under suitable hypotheses, we show that the resulting optimizers converge exponentially fast (and with exponentially decaying variance), with respect to increasing number of training samples. Numerical examples for optimal control, parameter identification and shape optimization problems for PDEs are provided to validate the proposed theory and to illustrate that ISMO significantly outperforms a standard deep neural network based surrogate optimization algorithm.

Praveen Chandrashekar

We propose a globally divergence conforming discontinuous Galerkin (DG) method on Cartesian meshes for {\em curl-type hyperbolic conservation} laws based on directly evolving the face and cell moments of the Raviart-Thomas approximation polynomials. The face moments are evolved using a 1-D discontinuous Gakerkin method that uses 1-D and multi-dimensional Riemann solvers while the cell moments are evolved using a standard 2-D DG scheme that uses 1-D Riemann solvers. The scheme can be implemented in a local manner without the need to solve a global mass matrix which makes it a truly DG method and hence useful for explicit time stepping schemes for hyperbolic problems. The scheme is also shown to exactly preserve the divergence of the vector field at the discrete level. Numerical results using second and third order schemes for induction equation are presented to demonstrate the stability, accuracy and divergence preservation property of the scheme.