NUMERICAL METHODS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS IN SCIENTIFIC SIMULATION AND ENGINEERING APPLICATIONS

Label-free learning of elliptic partial differential equation solvers with generalizability across boundary value problems

Flash memory-based computing-in-memory (CIM) architectures have gained popularity due to their remarkable performance in various computation tasks of data processing, including machine learning, neuron networks, and scientific calculations. Especially in the partial differential equation (PDE) solver that has been widely utilized in scientific calculations, high accuracy, processing speed, and low power consumption are the key requirements. This work proposes a novel flash memory-based PDE solver to implement PDE with high accuracy, low power consumption, and fast iterative convergence. Moreover, considering the increasing current noise in nanoscale devices, we investigate the robustness against the noise in the proposed PDE solver. The results show that the noise tolerance limit of the solver can reach more than five times that of the conventional Jacobi CIM solver. Overall, the proposed flash memory-based PDE solver offers a promising solution for scientific calculations that require high accuracy, low power consumption, and good noise immunity, which could help to develop flash-based general computing.

Most traditional methods for solving partial differential equations (PDEs) require the costly solving of large linear systems. Neural operators (NOs) offer remarkable speed-ups over classical numerical PDE solvers. Here, we conduct the first exploration and comparison of NOs for three-dimensional topology optimization. Specifically, we propose replacing the PDE solver within the popular Solid Isotropic Material with Penalization (SIMP) algorithm, which is its main computational bottleneck. For this, the NO not only needs to solve the PDE with sufficient accuracy but also has the additional challenge of providing accurate gradients which are necessary for SIMP’s density updates. To realize this, we do three things: (i) We introduce a novel loss term to promote gradient-consistency. (ii) We guarantee equivariance in our NOs to increase the physical correctness of predictions. (iii) We introduce a novel NO architecture called U-Net Fourier neural operator (U-Net FNO), which combines the multi-resolution properties of U-Nets with the Fourier neural operator (FNO)’s focus on local features in frequency space. In our experiments we demonstrate that the inclusion of the novel gradient loss term is necessary to obtain good results. Furthermore, enforcing group equivariance greatly improves the quality of predictions, especially on small training datasets. Finally, we show that in our experiments the U-Net FNO outperforms both a standard U-Net, as well as other FNO methods.

Image processing has largely been done via MATLAB and other programming languages, such as JAVA and C, implementing those Partial Differential Equation (PDE) models for denoising, smoothing, segmentation and restoration. This involves significant effort in writing implementation coding and in designing robust algorithms for image processing activities. In order to avoid such repeated effort such as algorithmic development for numerical solutions of PDE, we propose to use existing software packages which provide very good platforms of PDE solvers. Two commercial software platforms, namely PHYSICA3G and ANSYS, are examined. The former one allows a better interaction to the background coding and the latter one does not allow flexible user interaction with the background. This paper describes the implementation of PDE image processing models on and the design of a user interface with two very different software platforms, taking the advantage of the existing PDE solvers within them. The design logic is discussed in the paper along with numerical experiments on a number of methods, including isotropic model and anisotropic model for denoising and smoothing. Several pictures are used in the numerical experiments and the resulting PSNR ratios are obtained and compared with other work in the literature.

Comprehensive Coverage of the New, Easy-to-Learn C# Although C, C++, Java, and Fortran are well-established programming languages, the relatively new C# is much easier to use for solving complex scientific and engineering problems. Numerical Methods, Algorithms and Tools in C# presents a broad collection of practical, ready-to-use mathematical routines employing the exciting, easy-to-learn C# programming language from Microsoft. The book focuses on standard numerical methods, novel object-oriented techniques, and the latest Microsoft .NET programming environment. It covers complex number functions, data sorting and searching algorithms, bit manipulation, interpolation methods, numerical manipulation of linear algebraic equations, and numerical methods for calculating approximate solutions of non-linear equations. The author discusses alternative ways to obtain computer-generated pseudo-random numbers and real random numbers generated by naturally occurring physical phenomena. He also describes various methods for approximating integrals and special functions, routines for performing statistical analyses of data, and least squares and numerical curve fitting methods for analyzing experimental data, along with numerical methods for solving ordinary and partial differential equations. The final chapter offers optimization methods for the minimization or maximization of functions. Exploiting the useful features of C#, this book shows how to write efficient, mathematically intense object-oriented computer programs. The vast array of practical examples presented can be easily customized and implemented to solve complex engineering and scientific problems typically found in real-world computer applications.

Deep learning surrogate models aim to accelerate the solving of partial differential equations (PDEs) and have achieved certain promising results. Although several main-stream models through neural operator learning have been applied to delve into PDEs on varying geometries, they were designed to map the complex geometry to a latent uniform grid, which is still challenging to learn by the networks with general architectures. In this work, we rethink the critical factors of PDE solutions and propose a novel model-agnostic framework, called 3D Geometry-Guided Conditional adaptation (3D-GeoCA), for solving PDEs on arbitrary 3D geometries. Starting with a 3D point cloud geometry encoder, 3D-GeoCA can extract the essential and robust representations of any kind of geometric shapes, which conditionally guides the adaptation of hidden features in the surrogate model. We conduct experiments on two public computational fluid dynamics datasets, the Shape-Net Car and Ahmed-Body dataset, using several surrogate models as the backbones with various point cloud geometry encoders to simulate corresponding large-scale Reynolds Average Navier-Stokes equations. Equipped with 3D-GeoCA, these backbone models can reduce the L-2 error by a large margin. Moreover, this 3D-GeoCA is model-agnostic so that it can be applied to any surrogate model.

Coupling Mortar Finite Element and Boundary Element Methods for 2D Navier-Stokes Equations Courant Element: Before and After Iterative Methods for Solving Stiff Elliptic Problems Straight and Curved Finite Elements of Class C1 and Some Applications to Thin Shell Problems Exact Controllability to Solve the Helmholtz Equation with Absorbing Boundary Conditions Cubic Version of FEM in Elliptic Problems with Interfaces and Singularities Least Squares Mixed Finite Elements Turbulence Modelling in Finite Element Industrial Applications Necessary and Sufficient Conditions for the Numerical Approximation of a Partial Differential Equation Depending on a Small Parameter Efficient Solution Methods for Compressible Flow Computations Parallel Finite Volume Algorithms for Solving the Time-Domain Maxwell Equations on Nonstructured Meshes Coupling Between Nonlinear Maxwell and Heat Equations for an Induction Heating Problem: Modelling and Numerical Methods Solving the 3D Harmonic Maxwell Equations with Finite Elements, Lagrange Multipliers, and Iterative Methods Some Applications of the Hierarchic High Order MITC Finite Elements for Reissner-Mindlin Plates Domain Decomposition for Immiscible Displacement in Single Porosity Systems An Error Estimator for Nonconforming Approximations of a Nonlinear Problem Some Observations on Raviart-Thomas Mixed Finite Elements in p Extension for Parabolic Problems Mixed Finite Element Methods in Fluid Structure Systems A Black-Box Solver for the Solution of General Nonlinear Functional Equations by Mixed FEM A Remark on the Asymptotic Behaviour of Parabolic Variational Inequalities and Their Finite Element Approximation by the Courant Element Domain Decomposition vs. Adaptivity Material Optimization of Composites. Part Contents.

3D surface reconstruction from point clouds aims to generate continuous surface representations of physical objects from discrete spatial samples. This problem is of fundamental importance across diverse domains, including robotics, industrial design, and cultural heritage preservation. Among various surface representation schemes, Partial Differential equation (PDE)-based surfaces have emerged as a promising approach due to their strong mathematical foundations, compact representation, and high-order continuity properties. Despite these advantages, a key challenge lies in the analytical solution of PDEs used for surface modelling. Existing approaches often rely on numerical solvers, which are computationally expensive and prone to discretization errors. Although analytical solutions to fourth-order PDEs have been applied to parametric surface reconstruction, several limitations remain: (1) comparisons among different PDE orders (e.g., second-, fourth-, and sixth-order) have not been systematically explored; (2) hyperparameters within PDE solutions are typically tuned manually, which is inefficient and often suboptimal; (3) multiple analytical solutions to the same PDE are rarely compared in terms of their fitting accuracy; and (4) PDEs with non-zero sculpting force are rarely investigated for surface reconstruction. To address these issues, this study conducts a comprehensive analysis of closed-form PDE solutions of varying orders for parametric surface reconstruction from point clouds. The Genetic algorithm (GA) is employed to automatically determine optimal hyperparameter values, thereby improving reconstruction quality. Furthermore, we perform an extensive comparison among different PDE orders and among multiple analytical solutions derived from the same PDE. Lastly, we also investigate the impact of the sculpting force in the PDEs for surface reconstruction. Experimental results across several datasets of varying complexity demonstrate that the proposed framework achieves accurate and smooth surface reconstruction while significantly reducing manual intervention. This work provides new insights into the practical use of PDE-based parametric surfaces and establishes a foundation for future research in data-driven geometric modeling and surface reconstruction.

An introduction to scientific computing for differential equations Introduction to Computation and Modeling for Differential Equations provides a unified and integrated view of numerical analysis, mathematical modeling in applications, and programming to solve differential equations, which is essential in problem-solving across many disciplines, such as engineering, physics, and economics. This book successfully introduces readers to the subject through a unique Five-M approach: Modeling, Mathematics, Methods, MATLAB, and Multiphysics. This approach facilitates a thorough understanding of how models are created and preprocessed mathematically with scaling, classification, and approximation, and it also illustrates how a problem is solved numerically using the appropriate mathematical methods. The book's approach of solving a problem with mathematical, numerical, and programming tools is unique and covers a wide array of topics, from mathematical modeling to implementing a working computer program. The author utilizes the principles and applications of scientific computing to solve problems involving: Ordinary differential equations Numerical methods for Initial Value Problems (IVPs) Numerical methods for Boundary Value Problems (BVPs) Partial Differential Equations (PDEs) Numerical methods for parabolic, elliptic, and hyperbolic PDEs Mathematical modeling with differential equations Numerical solution Finite difference and finite element methods Real-world examples from scientific and engineering applications including mechanics, fluid dynamics, solid mechanics, chemical engineering, electromagnetic field theory, and control theory are solved through the use of MATLAB and the interactive scientific computing program Comsol Multiphysics. Numerous illustrations aid in the visualization of the solutions, and a related Web site features demonstrations, solutions to problems, MATLAB programs, and additional data. Introduction to Computation and Modeling for Differential Equations is an ideal text for courses in differential equations, ordinary differential equations, partial differential equations, and numerical methods at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for researchers and practitioners in the fields of mathematics, engineering, and computer science who would like to refresh and revive their knowledge of the mathematical and numerical aspects as well as the applications of scientific computation.

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

The main aim of this paper was to find the correct method of calculating equations of heat and mass transfer for the adsorption process and to calculate it numerically in reasonable time and with proper accuracy. An adsorption heat pump with a silica gel adsorbent and water adsorbate is discussed. We developed a mathematical model of temperature and uptake changes in the adsorber/desorber comprising the set of heat and mass balance partial differential equations (PDEs), together with the initial and boundary conditions and solved it by the numerical method of lines (NMOL). Spatial discretization was performed with equally spaced axial nodes and the PDEs were reduced to a set of ordinary differential equations (ODEs). We focused on the comparison of results obtained when the set of heat and mass balance ODEs for an adsorber was solved using: (1) the Runge–Kutta fixed step size fourth-order method (RKfixed), (2) the Runge–Kutta–Fehlberg 4.5th-order method with a variable step size (RK45), and (3) the Gear Backward Differentiation Formulae numerical (Gear BDF) methods. In our experience, all three types of ODE numerical methods (RKfixed, RK45, and Gear BDF) can be applied in simple models to model an adsorber with attention on their limitations. The Gear BDF method usually requires much fewer steps than the RK45 method for almost the same calculating time. RK methods require many more steps to obtain results, and the calculating time depends on accuracy or defined time step. Moreover, one should pay attention to the number of nodes or possible oscillations.

In prior-research the authors have demonstrated that, for stencil-based numerical solvers for Partial Differential Equations (PDEs), the parallel performance can be significantly improved by selecting sub-domains that are not cubic in shape (Saxena et. al., HPCS 2016, pp. 875-885). This is achieved through accounting for cache utilization in both the message passing and the computational kernel, where it is demonstrated that the optimal domain decompositions not only depend on the communication and load balance but also on the cache-misses, amongst other factors. In this work we demonstrate that those conclusions may also be extended to more advanced numerical discretizations, based upon Adaptive Mesh Refinement (AMR). In particular, we show that when basing our AMR strategy on the local refinement of patches of the mesh, the optimal patch shape is not typically cubic. We provide specific examples, with accompanying explanation, to show that communication minimizing strategies are not necessarily the best choice when applying AMR in parallel. All numerical tests undertaken in this work are based upon the open source BoxLib library.

Different numerical methods have been proposed for the solution of partial differential equations (PDE). Most of them are based on a variational principle which recasts the PDE into an equivalent integral equation. One of the most common principles is the Galerkin method, which has some specific disadvantages for some types of PDE. In this work an alternative variational principle, the least squares finite element method, will be tested with respect to its application for transient fluid-structure interaction problems. The accurracy of different formulations which were proposed for the Navier-Stokes equations in literature will be tested. In a next step these formulations will be coupled with a standard Galerkin approach for the structure. After that a new formulation for the linear equations of elastodynamics is developed and analysed with respect to its stability and accuracy. With this formulation it is possible do develop a pure least squares formulation for the strongly coupled fluid-structure problem. Finally the different formulations are tested with respect to their accuracy and efficiency.

Interface solutions of partial differential equations with point singularity

A numerical study of heat transfer and turbulent fluid flow over a staggered circular tube bank using a computational fluid dynamic code (Fluent 6.2) is presented in this work. The model for staggered arrangements is applicable for use over a wide range of the aspect ratio when determining heat transfer and turbulent fluid flow over a staggered circular tube bank. The turbulence model used involves the solution of two partial differential equations, one for the kinetic energy of the turbulence and the other for its dissipation rate. The results show that the average heat transfer for tube banks in cross flow depend on aspect ratio , Reynolds number, and Prandtl number. The present model gives higher heat transfer rates for all cases of Re number. The higher values of pressure drop distribution usually recorded near the initial of the external tube bank surface in all cases. The maximum velocity in the passage between the upper and lower side was recorded. The present results of the average Nusselt numbers with Re number at aspect ratio are in good agreement with the available data of Zukauskas and Ulinskas (1988). Therefore, the present values of the velocity vector at aspect ratio and Re = 20,000 were in good agreement with the analytical results of Ibrahim and Gomaa (2009).