We propose a particle method for numerically solving the Landau equation, inspired by the score-based transport modeling (SBTM) method for the Fokker-Planck equation. This method can preserve some important physical properties of the Landau equation, such as the conservation of mass, momentum, and energy, and decay of estimated entropy. We prove that matching the gradient of the logarithm of the approximate solution is enough to recover the true solution to the Landau equation with Maxwellian molecules. Several numerical experiments in low and moderately high dimensions are performed, with particular emphasis on comparing the proposed method with the traditional particle or blob method.
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In this paper, we propose and analyze a mixed formulation for the Kelvin-Voigt-Brinkman-Forchheimer equations for unsteady viscoelastic flows in porous media. Besides the velocity and pressure, our approach introduces the vorticity as a further unknown. Consequently, we obtain a three-field mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish the existence and uniqueness of a solution for the weak formulation, and derive the corresponding stability bounds, employing a fixed-point strategy, along with monotone operators theory and Schauder theorem. Afterwards, we introduce a semidiscrete continuous-in-time approximation based on stable Stokes elements for the velocity and pressure, and continuous piecewise polynomial spaces for the vorticity. Additionally, employing backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness, derive stability bounds, and establish the corresponding error estimates for both schemes. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.
The present work concerns the derivation of a numerical scheme to approximate weak solutions of the Euler equations with a gravitational source term. The designed scheme is proved to be fully well-balanced since it is able to exactly preserve all moving equilibrium solutions, as well as the corresponding steady solutions at rest obtained when the velocity vanishes. Moreover, the proposed scheme is entropy-preserving since it satisfies all fully discrete entropy inequalities. In addition, in order to satisfy the required admissibility of the approximate solutions, the positivity of both approximate density and pressure is established. Several numerical experiments attest the relevance of the developed numerical method.
Learning unknown stochastic differential equations (SDEs) from observed data is a significant and challenging task with applications in various fields. Current approaches often use neural networks to represent drift and diffusion functions, and construct likelihood-based loss by approximating the transition density to train these networks. However, these methods often rely on one-step stochastic numerical schemes, necessitating data with sufficiently high time resolution. In this paper, we introduce novel approximations to the transition density of the parameterized SDE: a Gaussian density approximation inspired by the random perturbation theory of dynamical systems, and its extension, the dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust density approximation, our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift and diffusion functions and computing the invariant distribution from trajectory data. And it is capable of handling trajectory data with low time resolution and variable, even uncontrollable, time step sizes, such as data generated from Gillespie's stochastic simulations. We then conduct several experiments across various scenarios to verify the advantages and robustness of the proposed method.
This paper presents a dissipativeness analysis of a quadrature method of moments (called HyQMOM) for the one-dimensional BGK equation. The method has exhibited its good performance in numerous applications. However, its mathematical foundation has not been clarified. Here we present an analytical proof of the strict hyperbolicity of the HyQMOM-induced moment closure systems by introducing a polynomial-based closure technique. As a byproduct, a class of numerical schemes for the HyQMOM system is shown to be realizability preserving under CFL-type conditions. We also show that the system preserves the dissipative properties of the kinetic equation by verifying a certain structural stability condition. The proof uses a newly introduced affine invariance and the homogeneity of the HyQMOM and heavily relies on the theory of orthogonal polynomials associated with realizable moments, in particular, the moments of the standard normal distribution.
We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.
Complex conjugate matrix equations (CCME) have aroused the interest of many researchers because of computations and antilinear systems. Existing research is dominated by its time-invariant solving methods, but lacks proposed theories for solving its time-variant version. Moreover, artificial neural networks are rarely studied for solving CCME. In this paper, starting with the earliest CCME, zeroing neural dynamics (ZND) is applied to solve its time-variant version. Firstly, the vectorization and Kronecker product in the complex field are defined uniformly. Secondly, Con-CZND1 model and Con-CZND2 model are proposed and theoretically prove convergence and effectiveness. Thirdly, three numerical experiments are designed to illustrate the effectiveness of the two models, compare their differences, highlight the significance of neural dynamics in the complex field, and refine the theory related to ZND.
Two arbitrary-order constraint-preserving schemes for the Yang--Mills equations on polyhedral meshes
Two numerical schemes are proposed and investigated for the Yang--Mills equations, which can be seen as a nonlinear generalisation of the Maxwell equations set on Lie algebra-valued functions, with similarities to certain formulations of General Relativity. Both schemes are built on the Discrete de Rham (DDR) method, and inherit from its main features: an arbitrary order of accuracy, and applicability to generic polyhedral meshes. They make use of the complex property of the DDR, together with a Lagrange-multiplier approach, to preserve, at the discrete level, a nonlinear constraint associated with the Yang--Mills equations. We also show that the schemes satisfy a discrete energy dissipation (the dissipation coming solely from the implicit time stepping). Issues around the practical implementations of the schemes are discussed; in particular, the assembly of the local contributions in a way that minimises the price we pay in dealing with nonlinear terms, in conjunction with the tensorisation coming from the Lie algebra. Numerical tests are provided using a manufactured solution, and show that both schemes display a convergence in $L^2$-norm of the potential and electrical fields in $\mathcal O(h^{k+1})$ (provided that the time step is of that order), where $k$ is the polynomial degree chosen for the DDR complex. We also numerically demonstrate the preservation of the constraint.
We present a new, monolithic first--order (both in time and space) BSSNOK formulation of the coupled Einstein--Euler equations. The entire system of hyperbolic PDEs is solved in a completely unified manner via one single numerical scheme applied to both the conservative sector of the matter part and to the first--order strictly non--conservative sector of the spacetime evolution. The coupling between matter and space-time is achieved via algebraic source terms. The numerical scheme used for the solution of the new monolithic first order formulation is a path-conservative central WENO (CWENO) finite difference scheme, with suitable insertions to account for the presence of the non--conservative terms. By solving several crucial tests of numerical general relativity, including a stable neutron star, Riemann problems in relativistic matter with shock waves and the stable long-time evolution of single and binary puncture black holes up and beyond the binary merger, we show that our new CWENO scheme, introduced two decades ago for the compressible Euler equations of gas dynamics, can be successfully applied also to numerical general relativity, solving all equations at the same time with one single numerical method. In the future the new monolithic approach proposed in this paper may become an attractive alternative to traditional methods that couple central finite difference schemes with Kreiss-Oliger dissipation for the space-time part with totally different TVD schemes for the matter evolution and which are currently the state of the art in the field.
This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial data, the regularity of the mild solution is investigated, and an error estimate is derived within the spatial (L^2)-norm setting. In the case of smooth initial data, two error estimates are established within the framework of general spatial (L^q)-norms.
We present a simple algorithm to approximate the viscosity solution of Hamilton-Jacobi~(HJ) equations by means of an artificial deep neural network. The algorithm uses a stochastic gradient descent-based algorithm to minimize the least square principle defined by a monotone, consistent numerical scheme. We analyze the least square principle's critical points and derive conditions that guarantee that any critical point approximates the sought viscosity solution. The use of a deep artificial neural network on a finite difference scheme lifts the restriction of conventional finite difference methods that rely on computing functions on a fixed grid. This feature makes it possible to solve HJ equations posed in higher dimensions where conventional methods are infeasible. We demonstrate the efficacy of our algorithm through numerical studies on various canonical HJ equations across different dimensions, showcasing its potential and versatility.
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