Advanced Topics in Computational Partial Differential Equations: Numerical Methods and Diffpack Programming

Solutions of Partial Differential Equations (PDEs) form the basis of many mathematical models in physics and medicine. In this work, a novel Tensor B-spline methodology for numerical solutions of linear second-order PDEs is proposed. The methodology applies the B-spline signal processing framework and computational tensor algebra in order to construct high-performance numerical solvers for PDEs. The method allows high-order approximations, is mesh-free, matrix-free and computationally and memory efficient. The first chapter introduces the main ideas of the Tensor B-spline method, depicts the main contributions of the thesis and outlines the thesis structure. The second chapter provides an introduction to PDEs, reviews the numerical methods for solving PDEs, introduces splines and signal processing techniques with B-splines, and describes tensors and the computational tensor algebra. The third chapter describes the principles of the Tensor B-spline methodology. The main aspects are 1) discretization of the PDE variational formulation via B-spline representation of the solution, the coefficients, and the source term, 2) introduction to the tensor B-spline kernels, 3) application of tensors and computational tensor algebra to the discretized variational formulation of the PDE, 4) tensor-based analysis of the problem structure, 5) derivation of the efficient computational techniques, and 6) efficient boundary processing and numerical integration procedures. The fourth chapter describes 1) different computational strategies of the Tensor B-spline solver and an evaluation of their performance, 2) the application of the method to the forward problem of the Optical Diffusion Tomography and an extensive comparison with the state-of-the-art Finite Element Method on synthetic and real medical data, 3) high-performance multicore CPU- and GPU-based implementations, and 4) the solution of large-scale problems on hardware with limited memory resources.

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Partial differential equations (PDEs) are fundamental in describing various physical phenomena, such as fluid dynamics, heat conduction, and wave propagation. However, analytical solutions to these equations are often difficult or impossible to obtain due to their complexity and the boundary conditions involved. Numerical methods provide an effective alternative by approximating solutions through discretization techniques. This paper explores various numerical methods for solving PDEs, including finite difference, finite element, and finite volume methods. We discuss their theoretical foundations, implementation strategies, and advantages in handling different types of PDEs, such as elliptic, parabolic, and hyperbolic equations. Moreover, the paper addresses key challenges such as stability, convergence, and computational efficiency, and reviews the use of high-performance computing in tackling large-scale problems. The applications of these methods in scientific computing and engineering are highlighted, demonstrating their versatility and importance in solving real-world problems.The numerical solution of partial differential equations (PDEs) plays a crucial role in solving real-world problems across various fields, including physics, engineering, and finance. Exact analytical solutions to PDEs are often not feasible due to their complexity and the nature of boundary conditions. As a result, numerical methods such as the finite difference, finite element, and finite volume methods are widely employed to approximate solutions. This paper provides an overview of these methods, emphasizing their formulation, implementation, and application to different types of PDEs, including elliptic, parabolic, and hyperbolic equations. Key considerations such as stability, convergence, and accuracy are discussed, along with strategies for improving computational efficiency. The paper also highlights the use of advanced computational techniques and parallel computing in addressing large-scale and complex PDE systems. Overall, numerical methods offer powerful tools for solving PDEs and are essential for simulating and analyzing complex phenomena in science and engineering.

11R1. Computational Partial Differential Equations: Numerical Methods and Diffpack Programming, Second Edition. - HP Langtangen (Simula Res Lab, Martin Linges vei 17, Fornebu, PO Box 134, Lysaker, 1325, Norway). Springer-Verlag, Berlin. 2003. 855 pp. ISBN 3-540-43416-X. $69.95.Reviewed by RL Huston (Dept of Mech, Indust, and Nucl Eng, Univ of Cincinnati, PO Box 210072, Cincinnati OH 45221-0072).This is the second edition of a popular tutorial on the numerical solution of partial differential equations (PDEs). It is intended for students, researchers, and practitioners interested in developing computer codes for the solution of the equations. The stated aim of the book is to equip the reader with skills for developing simulation software for physical phenomena (particularly, solid and fluid mechanics) governed by PDEs. The flow and style of the book are numeric together with listed computer codes. The software tools are based upon Diffpack–a numerical library using C++ and object oriented modules. Prior familiarity with C++ and Diffpack is thus obviously an advantage for potential readers. However, the book is written so that readers can learn both C++ and the use of Diffpack through a series of simple introductory examples and illustrations. The book is directed toward application in the various areas of solid and fluid mechanics. The book itself is divided into seven large chapters (or sections) together with four appendices spanning over 800 pages. Chapter 1 introduces the concepts of PDE solution using Diffpack. Elements of C++ programming are included. The chapter presents several illustrations of finite difference solution of the Poisson equation and the wave equation. Chapter 2 provides an introduction to the finite-element method starting with a discussion of weighted-residual methods and concluding with the mathematics of variational formulations. The third chapter presents a discussion of the use of Diffpack’s finite element software tools. Applications in heat transfer and the solution of the wave equation are given. Chapter 4 is devoted to nonlinear problems. It discusses discretation and the solution of nonlinear PDEs using both finite-difference and finite-element methods. Chapter 5, 6, and 7 present applications in solid mechanics, fluid mechanics, and coupled solid/fluid and fluid/heat transfer problems. The book concludes with four appendices providing extensive discussions of the underlying mathematics, Diffpack topics, linear systems, and software tools for solving linear systems. In the spirit of being a tutorial and text, Computational Partial Differential Equations: Numerical Methods and Diffpack Programming has over 150 exercises and a comparable number of worked-out examples together with computational code. There is an extensive bibliography of 156 references for further reading. The book is clearly very specialized but still devoted to an important aspect of applied mechanics. Therefore, it should be of interest and use to researchers and practitioners working in computational mechanics and to students aspiring to enter that field. It should make a good text for graduate-level numeric courses. Purchase by libraries is recommended.

The Feynman-Kac formula establishes a connection between stochastic processes and partial differential equations (PDEs), providing a novel approach for the numerical solution of PDEs by expressing the solution of a PDE as the expected value of a random variable. Moreover, this formula can be used to solve for the expected solution of stochastic partial differential equations (SPDEs). However, in practical applications, the formula is often constrained by the range of values that the random variable can take. When the values of the random variable are too large or too small, the numerical stability of the formula is reduced, and it may even become unusable. To address this issue, this paper proposes an adaptive algorithm that dynamically adjusts relevant parameters, enabling the formula to be applied to a broader range of situations. Furthermore, this adaptive algorithm is applied to high-dimensional stochastic partial differential equations, and a neural network algorithm is used to fit the expected solution of the SPDE. Experimental results show that, compared to traditional polynomial regression methods, this approach demonstrates higher precision and stability in high-dimensional problems. The adaptive algorithm proposed in this paper provides a novel approach for solving high-dimensional stochastic partial differential equations, with significant theoretical and practical value.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

In this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a conformal map and when mixed with the polynomial interpolation, it yields an accurate approximation. The first step to solve SPDE is to use stochastic Galerkin method in conjunction with polynomial chaos, which implies a system of deterministic partial differential equations to be solved. The main difficulty is the higher dimensionality of the resulting system of partial differential equations. The idea here is to solve this system using a small number of collocation points in space. This collocation technique is called Poly-Sinc and is used for the first time to solve high-dimensional systems of partial differential equations. Two examples are presented, mainly using Legendre polynomials for stochastic variables. These examples illustrate that we require to sample at few points to get a representation of a model that is sufficiently accurate.

Previous article Next article Reduction of the Number of Variables in Systems of Partial Differential Equations, with Auxiliary ConditionsM. J. Moran and R. A. GaggioliM. J. Moran and R. A. Gaggiolihttps://doi.org/10.1137/0116018PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Garrett Birkhoff, Hydrodynamics: A study in logic, fact and similitude, Revised ed, Princeton Univ. Press, Princeton, N.J., 1960xi+184 pp. (1 plate) MR0122193 0095.20303 Google Scholar[2] A. J. A. Morgan, The reduction by one of the number of independent variables in some systems of partial differential equations, Quart. J. Math., Oxford Ser. (2), 3 (1952), 250–259 MR0056183 0047.33403 CrossrefGoogle Scholar[3] Aristotle D. Michal, Differential invariants and invariant partial differential equations under continuous transformation groups in normed linear spaces, Proc. Nat. Acad. Sci. U. S. A., 37 (1951), 623–627 MR0043376 0054.04904 CrossrefISIGoogle Scholar[4] M. J. Moran, Masters Thesis, A unification of dimensional and similarity analysis via group theory, Doctoral dissertation, University of Wisconsin, Madison, 1967 Google Scholar[5] R. Manohar, Some similarity solutions of partial differential equations of boundary layer, Tech. Summary Rep., 375, Mathematics Research Center, University of Wisconsin, Madison, 1963 Google Scholar[6] Arthur G. Hansen, Similarity analyses of boundary value problems in engineering, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964xiv+114 MR0178596 0137.22603 Google Scholar[7] Luther Pfahler Eisenhart, Continuous groups of transformations, Dover Publications Inc., New York, 1961ix+301 MR0124008 0096.02103 Google Scholar[8] G. F. D. Duff, Partial differential equations, Mathematical expositions no. 9, University of Toronto Press, Toronto, 1956x+248 MR0078550 0071.30903 Google Scholar[9] R. A. Gaggioli and , M. J. Moran, Group theoretic techniques for the similarity solution of systems of partial differential equations with auxiliary conditions, Tech. Summary Rep., 693, Mathematics Research Center, University of Wisconsin, Madison, 1966 Google Scholar[10] Hermann Schlichting, Boundary layer theory, Translated by J. Kestin. 4th ed. McGraw-Hill Series in Mechanical Engineering, McGraw-Hill Book Co., Inc., New York, 1960xx+647 MR0122222 0096.20105 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Group method analysis for blood‐Mn‐ZnFe 2 O 4 flow and heat transfer under ferrohydrodynamics through a stretched cylinderMathematical Methods in the Applied Sciences, Vol. 28 | 21 June 2022 Cross Ref Extension of Blasius Newtonian Boundary Layer to Blasius Non-Newtonian Boundary LayerMathematical Journal of Interdisciplinary Sciences, Vol. 9, No. 2 | 8 June 2021 Cross Ref Forward scattering for non-linear wave propagation in (3 + 1)-dimensional Jimbo-Miwa equation using singular manifold and group transformation methodsWaves in Random and Complex Media, Vol. 9 | 21 July 2020 Cross Ref Advanced Ground Truth Multimodal Imaging Using Time Reversal (TR) Based Nonlinear Elastic Wave Spectroscopy (NEWS): Medical Imaging Trends Versus Non-destructive Testing ApplicationsRecent Advances in 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<i>Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems</i>

The theory of analytic systems of partial differential equations was first systematically investigated by Riquier and Elie Cartan around 1900. The existence of local solutions involves an algebraic problem, finding formal power series solutions, and an analytic problem, proving the convergence of formal power series solutions. Cartan defined the notion of an involutive system of partial differential equations and, using his theory of exterior differential systems, was able to show the existence of formal power series solutions for involutive partial differential equations of first order and to prove the convergence by the Cauchy-Kowalewski theorem. His result was extended by Kihler to systems of partial differential equations of higher order and is known today as the Cartan-Kihler theorem. Adjoining to a system of partial differential equations of order k the equations obtained by differentiating the original equations gives rise to a system of partial differential equations of order k +1, the prolongation of the system, which has the same solutions as the original equations. Cartan conjectured that, by prolonging a system a sufficient number of times, one would obtain an involutive system; this result was proved by Kuranishi in 1957 within the framework of Cartan's theory of exterior differential systems, and is now referred to as the Cartan-Kuranishi prolongation theorem. In 1961, Spencer introduced, in his fundamental paper [6] on the deformation of pseudogroup structures, certain cohomology groups Hkj associated to a partial differential equation (see ? 3), which are dual to homology groups of a Koszul complex; and so the cohomology groups Hki vanish for all sufficiently large k (see Lemma 3.1). The vanishing of these cohomology groups was shown by Serre to be equivalent to Cartan's notion of involutiveness (see V. W. Guillemin and S. Sternberg [3]). It then became possible to analyse the role played by involutiveness in Cartan's theory of partial differential equations. In this paper, we prove the Cartan-Kahler theorem for systems of linear partial differential equations formulated in terms of Ehresmann's theory of

Partial differential equations are paramount in mathematical modelling with applications in engineering and science. The book starts with a crash course on partial differential equations in order to familiarize the reader with fundamental properties such as existence, uniqueness and possibly existing maximum principles. The main topic of the book entails the description of classical numerical methods that are used to approximate the solution of partial differential equations. The focus is on discretization methods such as the finite difference, finite volume and finite element method. The manuscript also makes a short excursion to the solution of large sets of (non)linear algebraic equations that result after application of discretization method to partial differential equations. The book treats the construction of such discretization methods, as well as some error analysis, where it is noted that the error analysis for the finite element method is merely descriptive, rather than rigorous from a mathematical point of view. The last chapters focus on time integration issues for classical time-dependent partial differential equations. After reading the book, the reader should be able to derive finite element methods, to implement the methods and to judge whether the obtained approximations are consistent with the solution to the partial differential equations. The reader will also obtain these skills for the other classical discretization methods. Acquiring such fundamental knowledge will allow the reader to continue studying more advanced methods like meshfree methods, discontinuous Galerkin methods and spectral methods for the approximation of solutions to partial differential equations.

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.

Widely separated time-scales occur in many radio-frequency (RF) circuits, making the analysis with standard numerical methods difficult and costly. Low and high frequency signals are often superimposed enforcing very small time-steps over a long time-period in the computation of the numerical solution. Hence, classical numerical techniques result into long runtimes. In this paper we present a general method of embedding the underlying system of ordinary differential-algebraic equations (DAEs) in a system of partial differential equations (PDEs), such that a restriction of the solution of the PDEs onto a suitable path yields the desired solution of the DAEs. This allows to treat contributions with different frequencies separately in different dimensions, each dimension representing a time-scale. Along the coordinates the solution of the PDEs is typically very smooth, making numerical techniques for the solution of PDEs highly efficient. Here, theoretical results as well as new numerical methods are presented

Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. Some linear and most nonlinear differential equations are virtually impossible to solve using exact solutions, so it is often possible to find numerical or approximate solutions for such type of problems. Therefore, numerical methods are used to approximate the solution of such type of partial differential equation to the exact solution of partial differential equation. The finite-volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.

In this paper we studied the solution of partial differential equations using numerical methods. The paper includes study of the solving partial differential equations of the type of parabolic, elliptic and hyperbolic, and the method of the net was used for the numerical nods, which represents a case of finite differences. We have two types of solution which are the internal solution and boundary solution. The internal solution is based on the internal nodes of the net. The boundary solution depends on the boundary nodes of the net, in addition to finding the analytical solution of the equations to compare the results. We also discussed solving the problem of Laplace, Poisson, for the importance of these equations in the applied side; Mat lab was used to find the values of tables for the values of border differences. We have derived a new formula for the solution of partial differential equations containing three independent variables.