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Research Interests
My research interests have been on the numerical analysis, scientific
computing,
algorithm, partial differential equations and tangential suspension system.
I have been working on:
- Tangential Wheel Suspension System .
- Moving mesh finite element methods.
- Conservative front tracking.
- Back and forth error compensation and correction (BFECC) method with
applications in level set interface computation, fluid simulations, conveniently computing electromagnetic waves with complicated geometry etc.
A smoke simulation by NVIDIA using BFECC.
- Central schemes and central discontinuous Galerkin methods on overlapping cells for conservation laws
and associated differential equations.
- Non-oscillatory hierarchical reconstruction (HR) for discontinuous Galerkin methods,
central discontinuous Galerkin methods, central and finite volume schemes. A common mistake in the
implementation of HR..
- Neural Networks with Local Converging Inputs (NNLCI, preprint, publication) for solving differential equations in varying domains with orders of magnitude reduction in complexity and very small training data requirement. Predict a solution containing discontinuities, e.g., shocks, contacts and their interactions sharply and efficiently, see [1] (1D) and [2] (2D, 2023 featured article, Comm. in Comput. Phys.). Predict electromagnetic waves scattered off complicated perfect electric conductors (with training and prediction in different domains.) Predict supersonic flows in irregular domains with unstrucured grids (with training and prediction in different domains.) Predict solutions of a PNP ion channel model (nonlinear elliptic systems in multi domains.) The features of NNLCI include: 1. The neural network used can be small and simple. 2. A fine-grid simulation can provide hundreds or thousands of local samples for training. 3. The fine-grid simulations for training can be placed sparsely in the parameter space, thus covering more variations of the problem. 4. It can predict smooth solutions and discontinuities (such as shock interactions) accurately and sharply. 5. It is convenient to use on complex domain geometries, and the training and prediction can be done in different domains. 6. 2D numerical experiments show two orders of magnitude reduction in complexity for shock interactions, and about 500 time reductions in complexity for smooth solutions (electromagnetic waves.)
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