This paper presents a novel algorithm for solving incompressible Navier-Stokes equations, inspired by the Inverse Lax-Wendroff (ILW) boundary treatment. The ILW method leverages differential equations to compute spatial derivatives using time derivatives, thereby extending the solution to ghost points. We integrate the ILW computational process implicitly into the matrix assembly of our fluid-solving algorithm. To enhance the computational accuracy, we give an implementation of the ILW on the staggered grid. Theoretical analysis confirms the accuracy of our method, and numerical studies demonstrate its convergence and order of accuracy. Our approach realizes the simulation of viscous incompressible fluids within complex computational domains using a fixed Cartesian grid.
Efficient computation of elliptic eigenvalue problems is critical in science and engineering. In this paper, we propose a novel operator learning framework that directly maps arbitrary domains to their associated eigenvalues and eigenfunctions. Our approach employs a hybrid architecture combining Convolutional Neural Networks (CNNs) and Fourier Neural Operators (FNOs), enhanced by a robust preprocessing pipeline—including main axis alignment and detailed pixelization, that normalizes inputs to significantly simplify the learning task. Furthermore, we introduce a geometric-aware loss function specifically designed to accommodate domain variations and the inherent sign ambiguity of eigenfunctions, ensuring stable and consistent training. Extensive numerical experiments demonstrate that our method achieves superior accuracy and computational efficiency compared to traditional numerical solvers, particularly in many-query scenarios.
The Hadamard product of tensor train (TT) tensors is a fundamental nonlinear operation in scientific computing and data analysis. However, due to its tendency to significantly increase TT ranks, the Hadamard product poses a major computational challenge in TT-tensor-based algorithms. To address this, it is crucial to develop recompression algorithms that mitigate the effects of this rank increase. Existing recompression algorithms require an explicit representation of the Hadamard product, resulting in high computational and storage costs. In this work, we propose a Hadamard-avoiding TT recompression (HaTT) algorithm, which reduces both computational complexity and storage requirements. By leveraging the structure of the Hadamard product of TT tensors and exploiting its Hadamard product-free property, the HaTT algorithm achieves significantly lower complexity compared to existing TT recompression methods. This is confirmed through both complexity analysis and numerical experiments. Furthermore, the HaTT algorithm is applied to solve the Allen-Cahn equation, achieving substantial speedup over existing TT recompression algorithms without sacrificing accuracy.
Abstract. We present a conservation error analysis of non-conservative modified ghost fluid methods (MGFM) for axisymmetric compressible multi-medium flows within a discontinuous Galerkin framework. We propose a corresponding correction strategy. While multi-medium Riemann-solver-based MGFM-type methods (e.g., MGFM, MGFM) excel at simulating strong shock-interface interactions, their inherent non-conservatism can induce mass loss and pressure misalignment in long-time, high-energy-density computations. To address this issue, we theoretically analyze the conservation error introduced by the interface treatment, the cell-wise conservation error and the flux mismatch error. We prove that for MGFM-type methods, the global conservation error CE(M,N) is O(Δt). These errors are then conservatively redistributed to the grid cells adjacent to the interface, thereby eliminating the numerical losses of mass, momentum, and energy. The cell-wise error is redistributed using a local smoothing technique, while the flux mismatch error is allocated according to the wave propagation direction determined by the multi-material Riemann problem. The resulting algorithm integrates high-order discontinuous Galerkin discretizations with one-sided Riemann problems for the symmetry axis, forming a robust numerical framework for axisymmetric multi-medium flows in high-energy-density environments. Numerical experiments demonstrate that the proposed method not only achieves global conservation to machine precision but also effectively eliminates the pressure misalignment at the interface. In long-term, large-scale multi-medium simulations, the predicted key physical quantities are in excellent agreement with experimental data and reference solutions.
In high-temperature plasma physics, strong magnetic fields are essential for confining charged particles. Consequently, classical mathematical models of such systems must take into account the external magnetic field effects. A key governing equation is the magnetized Vlasov–Poisson system, which exhibits multiscale dynamics and rich physical properties. Thus, developing the structure-preserving numerical methods and studying its capability of maintaining these intrinsic properties over long-time simulations is therefore critically important. This paper presents a general framework for constructing and analyzing structure-preserving methods in orthogonal curvilinear coordinates. We prove that in these coordinates the Poisson-bracket structure is retained under appropriate finite element discretizations. However, the resulting Hamiltonian systems in transformed coordinates typically can’t be decomposed as several subsystems which can be solved exactly. To address this, we propose a semi-implicit numerical scheme that can still maintain the favorable stability properties inherited by the system. The effectiveness of the new derived numerical methods is demonstrated in application to strongly magnetized systems, and the rigorous asymptotic stability analysis are provided.
