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This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial values, the regularity of the mild solution is investigated, and an error estimate is derived with the spatial $ L^2 $-norm. For smooth initial values, two error estimates with the general spatial $ L^q $-norms are established.

A simplified kinetic description of rapid granular media leads to a nonlocal Vlasov-type equation with a convolution integral operator that is of the same form as the continuity equations for aggregation-diffusion macroscopic dynamics. While the singular behavior of these nonlinear continuity equations is well studied in the literature, the extension to the corresponding granular kinetic equation is highly nontrivial. The main question is whether the singularity formed in velocity direction will be enhanced or mitigated by the shear in phase space due to free transport. We present a preliminary study through a meticulous numerical investigation and heuristic arguments. We have numerically developed a structure-preserving method with adaptive mesh refinement that can effectively capture potential blow-up behavior in the solution for granular kinetic equations. We have analytically constructed a finite-time blow-up infinite mass solution and discussed how this can provide insights into the finite mass scenario.

In this paper, we introduce a framework for the discretization of a class of constrained Hamilton-Jacobi equations, a system coupling a Hamilton-Jacobi equation with a Lagrange multiplier determined by the constraint. The equation is non-local, and the constraint has bounded variations. We show that, under a set of general hypothesis, the approximation obtained with a finite-differences monotonic scheme, converges towards the viscosity solution of the constrained Hamilton-Jacobi equation. Constrained Hamilton-Jacobi equations often arise as the long time and small mutation asymptotics of population models in quantitative genetics. As an example, we detail the construction of a scheme for the limit of an integral Lotka-Volterra equation. We also construct and analyze an Asymptotic-Preserving (AP) scheme for the model outside of the asymptotics. We prove that it is stable along the transition towards the asymptotics. The theoretical analysis of the schemes is illustrated and discussed with numerical simulations. The AP scheme is also used to conjecture the asymptotic behavior of the integral Lotka-Volterra equation, when the environment varies in time.

A key numerical difficulty in compressible fluid dynamics is the formation of shock waves. Shock waves feature jump discontinuities in the velocity and density of the fluid and thus preclude the existence of classical solutions to the compressible Euler equations. Weak entropy solutions are commonly defined by viscous regularization, but even small amounts of viscosity can substantially change the long-term behavior of the solution. In this work, we propose the first inviscid regularization of the multidimensional Euler equation based on ideas from semidefinite programming, information geometry, geometric hydrodynamics, and nonlinear elasticity. From a Lagrangian perspective, shock formation in entropy solutions amounts to inelastic collisions of fluid particles. Their trajectories are akin to that of projected gradient descent on a feasible set of non-intersecting paths. We regularize these trajectories by replacing them with solution paths of interior point methods based on log determinantal barrier functions. These paths are geodesic curves with respect to the information geometry induced by the barrier function. Thus, our regularization replaces the Euclidean geometry of trajectories with a suitable information geometry. We extend this idea to infinite families of paths by viewing Euler's equations as a dynamical system on a diffeomorphism manifold. Our regularization embeds this manifold into an information geometric ambient space, equipping it with a geodesically complete geometry. Expressing the resulting Lagrangian equations in Eulerian form, we derive a regularized Euler equation in conservation form. Numerical experiments on one and two-dimensional problems show its promise as a numerical tool. While we focus on the barotropic Euler equations for concreteness and simplicity of exposition, our regularization easily extends to more general Euler and Navier-Stokes-type equations.

Acoustic wave equation is a partial differential equation (PDE) which describes propagation of acoustic waves through a material. In general, the solution to this PDE is nonunique. Therefore, initial conditions in the form of Cauchy conditions are imposed for obtaining a unique solution. Theoretically, solving the wave equation is equivalent to representing the wavefield in terms of a radiation source which possesses finite energy over space and time. In practice, the source may be represented in terms of pressure, normal derivative of pressure or normal velocity over a surface. The pressure wavefield is then calculated by solving an associated boundary value problem via imposing conditions on the boundary of a chosen solution space. From an analytic point of view, this manuscript aims to review typical approaches for obtaining unique solution to the acoustic wave equation in terms of either a volumetric radiation source $s$, or a singlet surface source in terms of normal derivative of pressure $(\partial/\partial \boldsymbol{n})p$ or its equivalent $\rho_0 u^{\boldsymbol{n}}$ with $\rho_0$ the ambient density, where $u^{\boldsymbol{n}} = \boldsymbol{u} \cdot \boldsymbol{n}$ is the normal velocity with $\boldsymbol{n}$ a unit vector outwardly normal to the surface. For some cases including a time-reversal propagation, the surface source is defined as a doublet source in terms of pressure $p$. A numerical approximation of the derived formulae will then be explained. The key step for numerically approximating the derived analytic formulae is inclusion of source, and will be studied carefully in this manuscript. It will be shown that compared to an analytical or ray-based solutions using Green's function, a numerical approximation of acoustic wave equation for a doublet source has a limitation regarding how to account for solid angles efficiently.

We give generators and relations for the hypergraph props of Gaussian relations and positive affine Lagrangian relations. The former extends Gaussian probabilistic processes by completely-uninformative priors, and the latter extends Gaussian quantum mechanics with infinitely-squeezed states. These presentations are given by adding a generator to the presentation of real affine relations and of real affine Lagrangian relations which freely codiscards effects, as well as certain rotations. The presentation of positive affine Lagrangian relations provides a rigorous justification for many common yet informal calculations in the quantum physics literature involving infinite-squeezing. Our presentation naturally extends Menicucci et al.'s graph-theoretic representation of Gaussian quantum states with a representation for Gaussian transformations. Using this graphical calculus, we also give a graphical proof of Braunstein and Kimble's continuous-variable quantum teleportation protocol. We also interpret the LOv-calculus, a diagrammatic calculus for reasoning about passive linear-optical quantum circuits in our graphical calculus. Moreover, we show how our presentation allows for additional optical operations such as active squeezing.

The Crouzeix--Raviart finite element method is widely recognized in the field of finite element analysis due to its nonconforming nature. The main goal of this paper is to present a general strategy for enhancing the Crouzeix--Raviart finite element using quadratic polynomial functions and three additional general degrees of freedom. To achieve this, we present a characterization result on the enriched degrees of freedom, enabling to define a new enriched finite element. This general approach is employed to introduce two distinct admissible families of enriched degrees of freedom. Numerical results demonstrate an enhancement in the accuracy of the proposed method when compared to the standard Crouzeix--Raviart finite element, confirming the effectiveness of the proposed enrichment strategy.

Time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$ are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness of their solutions. Two approaches are explored: the Lax-Milgram Theorem and the eigenfunction expansion. The resulting sufficient conditions, which involve certain $m\times m$ matrices (where $m$ is the order of the collocation scheme), are verified both analytically, for all $m\ge 1$ and all sets of collocation points, and computationally, for all $ m\le 20$. The semilinear case is also addressed.

In this paper we consider unconditionally energy stable numerical schemes for the nonstationary 3D magneto-micropolar equations that describes the microstructure of rigid microelements in electrically conducting fluid flow under some magnetic field. The first scheme is comprised of the Euler semi-implicit discretization in time and conforming finite element/stabilizedfinite element in space. The second one is based on Crank-Nicolson discretization in time and extrapolated treatment of the nonlinear terms such that skew-symmetry properties are retained. We prove that the proposed schemes are unconditionally energy stable. Some error estimates for the velocity field, the magnetic field, the micro-rotation field and the fluid pressure are obtained. Furthermore, we establish some first-order decoupled numerical schemes. Numerical tests are provided to check the theoretical rates and unconditionally energy stable.

Quantum computing has emerged as a promising avenue for achieving significant speedup, particularly in large-scale PDE simulations, compared to classical computing. One of the main quantum approaches involves utilizing Hamiltonian simulation, which is directly applicable only to Schr\"odinger-type equations. To address this limitation, Schr\"odingerisation techniques have been developed, employing the warped transformation to convert general linear PDEs into Schr\"odinger-type equations. However, despite the development of Schr\"odingerisation techniques, the explicit implementation of the corresponding quantum circuit for solving general PDEs remains to be designed. In this paper, we present detailed implementation of a quantum algorithm for general PDEs using Schr\"odingerisation techniques. We provide examples of the heat equation, and the advection equation approximated by the upwind scheme, to demonstrate the effectiveness of our approach. Complexity analysis is also carried out to demonstrate the quantum advantages of these algorithms in high dimensions over their classical counterparts.