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This work presents a suitable mathematical analysis to understand the properties of convergence and bounded variation of a new { fully discrete locally conservative} Lagrangian--Eulerian {explicit} numerical scheme to the entropy solution in the sense of Kruzhkov via weak asymptotic method. We also make use of the weak asymptotic method to connect the theoretical developments with the computational approach within the practical framework of a solid numerical analysis. This method also serves to address the issue of notions of solutions, and its resulting algorithms have been proven to be effective to study nonlinear wave formations and rarefaction interactions in intricate applications. The weak asymptotic solutions we compute in this study with our novel Lagrangian--Eulerian framework are shown to coincide with classical solutions and Kruzhkov entropy solutions in the scalar case. Moreover, we present and discuss significant computational aspects by means of numerical experiments related to nontrivial problems: a nonlocal traffic model, the $2 \times 2$ symmetric Keyfitz--Kranzer system, and numerical studies via Wasserstein distance to explain shock interaction with the fundamental inviscid Burgers' model for fluids. Therefore, the proposed weak asymptotic analysis, when applied to the Lagrangian--Eulerian framework, fits in properly with the classical theory while optimizing the mathematical computations for the construction of new accurate numerical schemes.

Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical properties, such as positivity or monotonicity. Such invalid solutions pose both modeling challenges, since the physical interpretation of simulation results is not possible, and computational challenges, since such properties may be required to advance the scheme. We, therefore, consider the problem of computing solutions that preserve these structural solution properties, which we enforce as additional constraints on the solution. We consider in particular the class of convex constraints, which includes positivity and monotonicity. By embedding such constraints as a postprocessing convex optimization procedure, we can compute solutions that satisfy general types of convex constraints. For certain types of constraints (including positivity and monotonicity), the optimization is a filter, i.e., a norm-decreasing operation. We provide a variety of tests on one-dimensional time-dependent PDEs that demonstrate the method's efficacy, and we empirically show that rates of convergence are unaffected by the inclusion of the constraints.

Runge-Kutta (RK) schemes, especially Gauss-Legendre and some other fully implicit RK (FIRK) schemes, are desirable for the time integration of parabolic partial differential equations due to their A-stability and high-order accuracy. However, it is significantly more challenging to construct optimal preconditioners for them compared to diagonally implicit RK (or DIRK) schemes. To address this challenge, we first introduce mathematically optimal preconditioners called block complex Schur decomposition (BCSD), block real Schur decomposition (BRSD), and block Jordan form (BJF), motivated by block-circulant preconditioners and Jordan form solution techniques for IRK. We then derive an efficient, near-optimal singly-diagonal approximate BRSD (SABRSD) by approximating the quasi-triangular matrix in real Schur decomposition using an optimized upper-triangular matrix with a single diagonal value. A desirable feature of SABRSD is that it has comparable memory requirements and factorization (or setup) cost as singly DIRK (SDIRK). We approximate the diagonal blocks in these preconditioning techniques using an incomplete factorization with (near) linear complexity, such as multilevel ILU, ILU(0), or a multigrid method with an ILU-based smoother. We apply the block preconditioners in right-preconditioned GMRES to solve the advection-diffusion equation in 3D using finite element and finite difference methods. We show that BCSD, BRSD, and BJF significantly outperform other preconditioners in terms of GMRES iterations, and SABRSD is competitive with them and the prior state of the art in terms of computational cost while requiring the least amount of memory.

We investigate the convergence of the Crouzeix-Raviart finite element method for variational problems with non-autonomous integrands that exhibit non-standard growth conditions. While conforming schemes fail due to the Lavrentiev gap phenomenon, we prove that the solution of the Crouzeix-Raviart scheme converges to a global minimiser. Numerical experiments illustrate the performance of the scheme and give additional analytical insights.

An energy conservative discontinuous Galerkin scheme for a generalised third order KdV type equation is designed. Based on the conservation principle, we propose techniques that allow for the derivation of optimal a priori bounds for the linear KdV equation and a posteriori bounds for the linear and modified KdV equation. Extensive numerical experiments showcasing the good long time behaviour of the scheme are summarised which are in agreement with the analysis proposed.

We introduce a general framework for enforcing local or global maximum principles in high-order space-time discretizations of a scalar hyperbolic conservation law. We begin with sufficient conditions for a space discretization to be bound preserving (BP) and satisfy a semi-discrete maximum principle. Next, we propose a global monolithic convex (GMC) flux limiter which has the structure of a flux-corrected transport (FCT) algorithm but is applicable to spatial semi-discretizations and ensures the BP property of the fully discrete scheme for strong stability preserving (SSP) Runge-Kutta time discretizations. To circumvent the order barrier for SSP time integrators, we constrain the intermediate stages and/or the final stage of a general high-order RK method using GMC-type limiters. In this work, our theoretical and numerical studies are restricted to explicit schemes which are provably BP for sufficiently small time steps. The new GMC limiting framework offers the possibility of relaxing the bounds of inequality constraints to achieve higher accuracy at the cost of more stringent time step restrictions. The ability of the presented limiters to preserve global bounds and recognize well-resolved smooth solutions is verified numerically for three representative RK methods combined with weighted essentially nonoscillatory (WENO) finite volume space discretizations of linear and nonlinear test problems in 1D.

The main aim of this study is to introduce a 2-layered Artificial Neural Network (ANN) for solving the Black-Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of Ordinary Differential Equations (ODE). Then each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine tuning for speeding up the process and domain mapping to confront infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of Black-Scholes model are reported.

We revisited the relation between the gradient equations and Hamilton's equations in information geometry. By regarding a gradient-flow equation in information geometry as Huygens' equation in geometric optics, we relate the gradient flow to the geodesic flow induced by a geodesic Hamiltonian in Riemannian geometry. The original time parameter $t$ in the gradient equations is related to the arc-length parameter in the Riemannian manifold by Jacobi-Maupertuis transformation. As a by-product, it is found the relation between the gradient equation and replicator equations.

We propose a geometric structure induced by any given convex polygon $P$, called $Nest(P)$, which is an arrangement of $\Theta(n^2)$ line segments, each of which is parallel to an edge of $P$, where $n$ denotes the number of edges of $P$. We then deduce six nontrivial properties of $Nest(P)$ from the convexity of $P$ and the parallelism of the line segments in $Nest(P)$. Among others, we show that $Nest(P)$ is a subdivision of the exterior of $P$, and its inner boundary interleaves the boundary of $P$. They manifest that $Nest(P)$ has a surprisingly good interaction with the boundary of $P$. Furthermore, we study some computational problems on $Nest(P)$. In particular, we consider three kinds of location queries on $Nest(P)$ and answer each of them in (amortized) $O(\log^2n)$ time. Our algorithm for answering these queries avoids an explicit construction of $Nest(P)$, which would take $\Omega(n^2)$ time. By applying the aforementioned six properties altogether, we find that the geometric optimization problem of finding the maximum area parallelogram(s) in $P$ can be reduced to answering $O(n)$ aforementioned location queries, and thus be solved in $O(n\log^2n)$ time. This application will be reported in a subsequent paper.