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1. Positivity-preserving and energy-dissipative finite difference schemes for the Fokker–Planck and Keller–Segel equations

   https://doi.org/10.1093/imanum/drac014  

   Hu, Jingwei; Zhang, Xiangxiong (May 2022, IMA Journal of Numerical Analysis)

   Abstract In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker–Planck equation and the Keller–Segel equations. The two proposed schemes are first-order accurate in time, explicitly solvable, and second-order and fourth-order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proved to be positivity preserving and energy dissipative: the second-order scheme can achieve so unconditionally while the fourth-order scheme only requires a mild time step and mesh size constraint. In particular, the fourth-order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions. 

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2. HoD-Net: High-Order Differentiable Deep Neural Networks and Applications

   https://doi.org/10.1609/aaai.v36i8.20799  

   Shen, Siyuan; Shao, Tianjia; Zhou, Kun; Jiang, Chenfanfu; Luo, Feng; Yang, Yin (June 2022, Proceedings of the AAAI Conference on Artificial Intelligence)

   We introduce a deep architecture named HoD-Net to enable high-order differentiability for deep learning. HoD-Net is based on and generalizes the complex-step finite difference (CSFD) method. While similar to classic finite difference, CSFD approaches the derivative of a function from a higher-dimension complex domain, leading to highly accurate and robust differentiation computation without numerical stability issues. This method can be coupled with backpropagation and adjoint perturbation methods for an efficient calculation of high-order derivatives. We show how this numerical scheme can be leveraged in challenging deep learning problems, such as high-order network training, deep learning-based physics simulation, and neural differential equations. 

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3. High-precision quantum algorithms for partial differential equations

   https://doi.org/10.22331/q-2021-11-10-574  

   Childs, Andrew M.; Liu, Jin-Peng; Ostrander, Aaron (November 2021, Quantum)

   Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity p o l y ( 1 / ϵ ) , where ϵ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be p o l y ( d , log ⁡ ( 1 / ϵ ) ) , where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations. 

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4. Numerical methods and improvements for simulating quasi-static elastoplastic materials

   https://doi.org/10.1016/j.jcp.2025.113756  

   Lu, Jiayin; Rycroft, Chris H (March 2025, Journal of Computational Physics)

   Hypo-elastoplasticity is a framework suitable for modeling the mechanics of many hard materials that have small elastic deformation and large plastic deformation. In laboratory tests for these materials the Cauchy stress is often in quasi-static equilibrium. Rycroft et al. discovered a mathematical correspondence between this physical system and the incompressible Navier–Stokes equations, and developed a projection method similar to Chorin's projection method (1968) for incompressible Newtonian fluids. Here, we improve the original projection method to simulate quasi-static hypo-elastoplasticity, by making three improvements. First, drawing inspiration from the second-order projection method for incompressible Newtonian fluids, we formulate a second-order in time numerical scheme for quasi-static hypo-elastoplasticity. Second, we implement a finite element method for solving the elliptic equations in the projection step, which provides both numerical benefits and flexibility. Third, we develop an adaptive global time-stepping scheme, which can compute accurate solutions in fewer timesteps. Our numerical tests use an example physical model of a bulk metallic glass based on the shear transformation zone theory, but the numerical methods can be applied to any elastoplastic material. 

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5. Finite element approximation of an obstacle problem for a class of integro–differential operators

   https://doi.org/10.1051/m2an/2019058  

   Bonito, Andrea; Lei, Wenyu; Salgado, Abner J. (January 2020, ESAIM: Mathematical Modelling and Numerical Analysis)

   We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme. 

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