A mesh free method using rectangular pre-solved domains using Kronecker products

ABSTRACTIn the recent project BENCHOP – the BENCHmarking project in Option Pricing we found that Stochastic and Local Volatility problems were particularly challenging. Here we continue the effort by introducing a set of benchmark problems for this type of problems. Eight different methods targeted for the Stochastic Differential Equation (SDE) formulation and the Partial Differential Equation (PDE) formulation of the problem, as well as Fourier methods making use of the characteristic function, were implemented to solve these problems. Comparisons are made with respect to time to reach a certain error level in the computed solution for the different methods. The implemented Fourier method was superior to all others for the two problems where it was implemented. Generally, methods targeting the PDE formulation of the problem outperformed the methods for the SDE formulation. Among the methods for the PDE formulation the ADI method stood out as the best performing one.

We consider the collocation method for linear, second-order elliptic problems on rectangular and general two-dimensional domains. An overview of the method is given for general domains, followed by a discussion of the improved efficiencies and simplifications possible for rectangular domains. A very-high-level description is given of three specific collocation algorithms that use Hermite bicubic basic functions, (1) GENCOL (collocation on general two-dimensional domains), (2) HERMCOL (collocation on rectangular domains with general linear boundary conditions), and (3) INTCOL (collocation on rectangular domains with uncoupled boundary conditions). The linear system resulting from INTCOL has half the bandwidth of that from HERMCOL, which provides substantial benefit in solving the system. We provide some examples showing the range of applicability of the algorithms and some performance profiles illustrating their efficiency. Fortran implementations of these algorithms are given in the companion papers [10, 11].

Currently, nanofluid is a hot area of interest for researchers. The nanofluid with bioconvection phenomenon attracted the researchers owing to its numerous applications in the field of nanotechnology, microbiology, nuclear science, heat storage devices, biosensors, biotechnology, hydrogen bomb, engine of motors, cancer treatment, the atomic reactor, cooling of devices, and in many more. This article presents the bioconvection cross-diffusion effects on the magnetohydrodynamic flow of nanofluids on three different geometries (cone, wedge, and plate) with mixed convection. The temperature-dependent thermal conductivity, thermal diffusivity, and Arrhenius activation energy applications are considered on the fluid flow with melting phenomenon. The flow is analyzed under thermal and solutal Robin’s conditions. The problem is formulated in the mathematical formulation of partial differential equations (PDEs). The similarity transformations are applied to diminish the governing nonlinear coupled boundary value problems into higher-order non-linear ordinary differential equations (ODEs). The resulting expressions/equation numerically tackled utilizing the famous bvp4c package by MATLAB for various interesting parameters. The results were physically and numerically calculated through graphics and tables for the velocity field, energy distribution, nanoparticles concentration, and microorganisms profile for numerous parameters. From the obtained results, we discern that the transfer of heat and mass coefficient is high over a plate and cone in the flow, respectively. The velocity profile is reduced via a larger magnetic parameter. Temperaturedependent thermal conductivity enhances the thermal field. Larger thermophoresis enhanced the concentration of nanoparticles. The microorganisms’ Biot number boosts the microorganism’s profile.

Interface solutions of partial differential equations with point singularity

In this work we study the application of the Method of Fundamental Solutions (MFS) to the numerical calculation of the eigenvalues and eigenfunctions for the 2D Bilaplacian in simply connected plates. This problem was considered in [5] using wave-type functions and in [4] using radial basis function for a circular and rectangular domain. The MFS is a meshfree method that was already applied to the calculation of the eigenvalues and eigenfunctions associated to the Laplace operator (cf. [6], [2] and [1]). The application of this method to the Bilaplacian was already considered in [3], but only for simple shapes. Here we apply an algorithm for the choice of point-sources, as in [1], that leads to very good results for a fairly general class of domains.

Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adaptive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically.

A novel space–time meshfree collocation method (STMCM) for solving systems of non‐linear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh‐based methods. The STMCM belongs to the class of truly meshfree methods, i.e. the methods that do not have any underlying mesh, but work on a set of nodes only without any a priori node‐to‐node connectivity. Instead, the neighbouring information is established on‐the‐fly. The STMCM is constructed using the Interpolating Moving Least‐squares technique, which allows a simplified implementation of boundary conditions due to fulfillment of the Kronecker delta property by the kernel functions, which is not the case for the major part of other meshfree methods.The method is validated by several examples ranging from interpolation problems to the solution of PDEs, whereas the STMCM solutions are compared with either analytical or reference ones. Copyright © 2009 John Wiley & Sons, Ltd.

This paper presents a new parallel domain decomposition algorithm based on integer linear programming (ILP), a mathematical optimization method. To minimize the computation time of coastal ocean circulation models, the ILP decomposition algorithm divides the global domain in local domains with balanced work load according to the number of processors and avoids computations over as many as land grid cells as possible. In addition, it maintains the use of logically rectangular local domains and achieves the exact same results as traditional domain decomposition algorithms (such as Cartesian decomposition). However, the ILP decomposition algorithm may not converge to an exact solution for relatively large domains. To overcome this problem, we developed two ILP decomposition formulations. The first one (complete formulation) has no additional restriction, although it is impractical for large global domains. The second one (feasible) imposes local domains with the same dimensions and looks for the feasibility of such decomposition, which allows much larger global domains. Parallel performance of both ILP formulations is compared to a base Cartesian decomposition by simulating two cases with the newly created parallel version of the Stevens Institute of Technology’s Estuarine and Coastal Ocean Model (sECOM). Simulations with the ILP formulations run always faster than the ones with the base decomposition, and the complete formulation is better than the feasible one when it is applicable. In addition, parallel efficiency with the ILP decomposition may be greater than one.

This investigation delves into the impact of double diffusion on the heat-generating hydro-magnetic transport of water-based nanofluids flowing past a moving permeable plate subjected to heat flux. The nanofluids considered incorporate copper (Cu) and titanium dioxide (TiO₂) nanoparticles. The study is characterized by the formulation of dimensional quasi-linear partial differential equations (PDEs) with well-defined initial and boundary conditions, which are rendered dimensionless by introducing appropriate non-dimensional parameters. These equations are then resolved numerically using the implicit Crank-Nicolson finite difference scheme implemented in MATLAB. The research meticulously examines the influence of pertinent physical parameters on the velocity profile, temperature distribution, and concentration of reactive species, complemented by graphical illustrations. Furthermore, the study elucidates the impact of these physical variables on the wall shear stress, heat transfer rate, and mass transfer rate, presented in tabular format and subjected to comprehensive analysis. The findings underscore that an increase in the Soret parameter enhances fluid velocity and reactive species concentration, whereas the intensification of the magnetic field and heat flux parameter diminishes fluid velocity. The rate of thermal transfer is observed to escalate with higher values of the heat source, radiation-absorption, and radiative heat flux parameters; conversely, it diminishes with an increase in the Prandtl number. The study reveals superior thermal performance for Cu nanoparticles compared to TiO₂. A comparative analysis with prior published results substantiates the accuracy of the computational framework, with the outcomes exhibiting exceptional concordance.

A generalization of a NURBS based parametric mesh-free method (NPMM), recently proposed by Shaw and Roy (2008), is considered. A key feature of this parametric formulation is a geometric map that provides a local bijection between the physical domain and a rectangular parametric domain. This enables constructions of shape functions and their derivatives over the parametric domain whilst satisfying polynomial reproduction and interpolation properties over the (non-rectangular) physical domain. Hence the NPMM enables higher-dimensional B-spline based functional approximations over non-rectangular domains even as the NURBS basis functions are constructed via the usual tensor products of their one-dimensional counterparts. Nevertheless the method still lacks the universality that the FEM enjoys. In particular, for many non-simply connected domains, the geometric map may not be locally bijective everywhere and this severely restricts the general applicability of the NPMM. In this paper, a piecewise form of the NPMM is proposed, wherein the domain is decomposed into a collection of simply connected sub-domains or element patches (analogous to the FEM). The NPMM is then employed over each sub-domain without affecting the continuity of approximated functions across inter-sub-domain boundaries. This is quite unlike the usual FEM. The proposed procedure not only possesses the generality of the FEM, it is also equipped with higher order, globally smooth and interpolating basis functions. It may thus be interpreted as a seamless bridge between the FEM and mesh-free methods. In the context of weak implementations of the piecewise NPMM, we propose a conformal knot-grid integration scheme. Finally, we illustrate these schemes for weak numerical solutions of a few linear and nonlinear boundary value problems of engineering interest.

Error estimates for a class of partial functional differential equation with small dissipation

In this paper we consider the Lagrangian formulation of a system of second order quasilinear partial differential equations. Specifically we construct a Lagrangian vector field such that the flows of the vector field satisfy the original system of partial differential equations.

The adaptive wavelet Galerkin method for solving linear, elliptic operator equations introduced by Cohen et al. (Math Comp 70:27---75, 2001) is extended to nonlinear equations and is shown to converge with optimal rates without coarsening. Moreover, when an appropriate scheme is available for the approximate evaluation of residuals, the method is shown to have asymptotically optimal computational complexity. The application of this method to solving least-squares formulations of operator equations $$G(u)=0$$G(u)=0, where $$G:H \rightarrow K'$$G:HźKź, is studied. For formulations of partial differential equations as first-order least-squares systems, a valid approximate residual evaluation is developed that is easy to implement and quantitatively efficient.

In this chapter we define Sobolev spaces which are the ‘natural’ spaces of functions in which to solve variational formulations of partial differential equations. Physically, Sobolev spaces can be interpreted as spaces of functions with finite energy. This chapter is the most ‘technical’ of this book and relies in part on a course of ‘pure’ mathematics Nevertheless, it is necessary to understand the results below well to follow the rest, of the course, including its more numerical aspects. In general, it is not necessary to know the proofs of these results (expect for the most simple and most useful).

Gliomas are the most dominant and lethal type of brain tumors. Growth prediction is significant to quantify tumor aggressiveness, improve therapy planning, and estimate patients' survival time. This is commonly addressed in literature using mathematical models guided by multi-time point scans of multi/single-modal data for the same subject. However, these models are mechanism-based and heavily rely on complicated mathematical formulations of partial differential equations with few parameters that are insufficient to capture different patterns and other characteristics of gliomas. In this paper, we propose a 3D generative adversarial networks (GANs) for glioma growth prediction. Specifically, we stack 2 GANs with conditional initialization of segmented feature maps. Furthermore, we employ Dice loss in our objective function and devised 3D U-Net architecture for better image generation. The proposed method is trained and validated using 3D patch-based strategy on real magnetic resonance images of 9 subjects with 3 time points. Experimental results show that the proposed method can be successfully used for glioma growth prediction with satisfactory performance.