Partial Differential Equation Solver Research Articles

ReLOPE: Resistive RAM-Based Linear First-Order Partial Differential Equation Solver

Data movement between memory and processing units poses an energy barrier to Von-Neumann-based architectures. In-memory computing (IMC) eliminates this barrier. RRAM-based IMC has been explored for data-intensive applications, such as artificial neural networks and matrix-vector multiplications that are considered as “soft” tasks where performance is a more important factor than accuracy. In “hard” tasks such as partial differential equations (PDEs), accuracy is a determining factor. In this brief, we propose ReLOPE, a fully RRAM crossbar-based IMC to solve PDEs using the Runge–Kutta numerical method with 97% accuracy. ReLOPE expands the operating range of solution by exploiting shifters to shift input data and output data. ReLOPE range of operation and accuracy can be expanded by using fine-grained step sizes by programming other RRAMs on the BL. Compared to software-based PDE solvers, ReLOPE gains $31.4\times $ energy reduction at only 3% accuracy loss.

Gas discharge mechanism of weakly ionized gas sensor using nanomaterials

Gas discharge plays a very important role in the performance of ionized gas sensor. Due to the structure of nanomaterial-based electrode, gas discharge becomes more complexity under atmospheric pressure. In this article, we show a theoretical framework for the calculation of gas discharge of weakly ionized gas sensor using nanomaterials. It consists of the mass conservation equation for each species with the drift-diffusion approximation, the electron energy conservation equation, and the Poisson’s equation. The effects of ionization, excitation, attachment, recombination, electron diffusion, and is formulated by a time-dependent finite-element partial differential equation solver. The simulation results agree well with experimental data under the same conditions and gas discharge characters of ionized gas sensor can be extracted from the model to understand the discharge mechanism. Our approach is not only suitable for explaining the experimental phenomena by clarifying the gas chemical reaction process but also to design the sensor structure.

Solving Fundamental Solution of Non-Homogeneous Heat Equation with Dirichlet Boundary Conditions

In this study, we developed a solution of nonhomogeneous heat equation with Dirichlet boundary conditions. moreover, the non-homogeneous heat equation with constant coefficient. since heat equation has a simple form, we would like to start from the heat equation to find the exact solution of the partial differential equation with constant coefficient. to emphasize our main results, we also consider some important way of solving of partial differential equation specially solving heat equation with Dirichlet boundary conditions. the main results of our paper are quite general in nature and yield some interesting solution of non-homogeneous heat equation with Dirichlet boundary conditions and it is used for problems of mathematical modeling and mathematical physics.

Development of a Massively Parallelized Fluid-Based Plasma Simulation Code With a Finite-Volume Method on an Unstructured Grid

A massively parallelized unstructured-grid plasma simulation code based on a multifluid plasma model has been developed and validated. The code uses a collocated cell-centered finite-volume method and is designed to simulate intermediate low-pressure and atmospheric-pressure (AP) plasma sources with complex machine geometries. One of the novel features of this code is that it is implemented in a highly flexible computational platform named ultrafast massively parallel processing (ultraMPP), which allows straightforward addition and integration of different partial differential equation (PDE) solvers in a self-consistent manner. As to the numerical methods, the Scharfetter–Gummel scheme is adopted to handle the drift-diffusion flux of electrons, whereas the Harten–Lax-van Leer (HLL)-type approximate Riemann solver is used to handle the convection terms of ion momentum equations. For discretization of the diffusion terms in an unstructured grid, the Taylor expansion is used to deal with the effects of nonorthogonality of the cells, and the cell-center gradient is calculated using a least-squares method. The simulation code with a local-field approximation (LFA) was validated for two cases of AP plasmas as well as a case of an argon capacitively coupled plasma generated in a Gaseous Electronic Conference (GEC) reference cell with local mean energy approximation (LMEA) at intermediate low pressure. The simulation results were found to be in good agreement with the previously published experimental and simulation data.

PDE-NetGen 1.0: from symbolic partial differential equation (PDE) representations of physical processes to trainable neural network representations

Abstract. Bridging physics and deep learning is a topical challenge. While deep learning frameworks open avenues in physical science, the design of physically consistent deep neural network architectures is an open issue. In the spirit of physics-informed neural networks (NNs), the PDE-NetGen package provides new means to automatically translate physical equations, given as partial differential equations (PDEs), into neural network architectures. PDE-NetGen combines symbolic calculus and a neural network generator. The latter exploits NN-based implementations of PDE solvers using Keras. With some knowledge of a problem, PDE-NetGen is a plug-and-play tool to generate physics-informed NN architectures. They provide computationally efficient yet compact representations to address a variety of issues, including, among others, adjoint derivation, model calibration, forecasting and data assimilation as well as uncertainty quantification. As an illustration, the workflow is first presented for the 2D diffusion equation, then applied to the data-driven and physics-informed identification of uncertainty dynamics for the Burgers equation.

Scalability of high-performance PDE solvers

Performance tests and analyses are critical to effective high-performance computing software development and are central components in the design and implementation of computational algorithms for achieving faster simulations on existing and future computing architectures for large-scale application problems. In this article, we explore performance and space-time trade-offs for important compute-intensive kernels of large-scale numerical solvers for partial differential equations (PDEs) that govern a wide range of physical applications. We consider a sequence of PDE-motivated bake-off problems designed to establish best practices for efficient high-order simulations across a variety of codes and platforms. We measure peak performance (degrees of freedom per second) on a fixed number of nodes and identify effective code optimization strategies for each architecture. In addition to peak performance, we identify the minimum time to solution at 80% parallel efficiency. The performance analysis is based on spectral and p-type finite elements but is equally applicable to a broad spectrum of numerical PDE discretizations, including finite difference, finite volume, and h-type finite elements.

A robust and accurate solver for some nonlinear partial differential equations and tow applications

In this paper we introduce a robust and accurate solver in order to solve various classes of nonlinear partial differential equations. This solver gives the closed formula for the solutions. Within this respect, a solver is developed to accurately resolve and represent the complete wave structure of the nonlinear partial differential equations. To verify this solver, the two applications are introduced. The obtained solutions may be applicable for some new observations in physics, fluid mechanics, biology, engineering, economy, astrophysics, meteorological topics, gravitational collapse and laser technology. The theoretical study and obtained solutions reported illustrating that the proposed solver efficient and adequate. The shape for the obtained solutions are illustrated via 2D and 3D plots for suitable values of the free parameters.

OpenCAL system extension and application to the three-dimensional Richards equation for unsaturated flow

OpenCAL is a scientific software library specifically developed for the simulation of 2D and 3D complex dynamical systems on parallel computational devices. It is written in C/C++ and relies on OpenMP/OpenCL and MPI for parallel execution on multi-/many-core devices and clusters of computers, respectively. The library provides the Extended Cellular Automata paradigm as a high-level formalism for modeling complex systems on structured computational grids. As a consequence, it can be used as a parallel explicit solver of ordinary and partial differential equations through classical methods, including finite-difference and finite-volume. Here the latest version of the library is described, introducing the MPI infrastructure over the 3D OpenCL and 2D/3D OpenMP components. The implementation of a three-dimensional unsaturated flow model based on a direct discrete formulation of the Richards’ equation is also shown, corresponding to a finite-difference scheme. Computational performances have been assessed on both a scientific workstation equipped with a dual Intel Xeon socket and three Nvidia GPUs, and a 16 nodes cluster with a fast interconnection network. The OpenCAL embedded quantization optimization is also discussed and exploited to drastically reduce computing time.

Phononic transport and simulations of annealing processes in nanometric complex structures

Modeling thermal transport at the nanoscale is a difficult task, especially when external time-varying heating sources and the complexity of the studied systems make computational schemes that rely on accurate particlelike simulations nonaffordable. Alternative strategies based on corrections of the Fourier law could satisfy the trade-off between accuracy and computational efficiency, since they can be implemented in partial differential equation solvers. This continuum approach could also allow for the coupling between thermal transport and other evolving fields related to the generalized temperature field. Here we demonstrate that corrections due to the finite phonon mean free paths can be suitably included in annealing process simulations of three-dimensional nanosystems. Quantitative predictions can be obtained and readily compared with the experimental characterization of the processed samples.

Two‐phase natural convection analysis and hybrid nanoparticle migration around micromixer blades

AbstractIn this paper, numerical investigations are presented for hybrid nanoparticle migration and free convection heat transfer of two kinds of nanofluids in a micromixer at the fixed propeller condition. The inner blades and outer crust of the micromixer are kept at constant hot and cold temperatures, respectively. Two kinds of hybrid nanofluids, TiO2‐CuO water and ethylene glycol‐(MoS2‐SiO2), are considered. The governing equations including velocity, pressure, temperature formulation, and nanoparticle concentrations are solved by a partial differential equation solver based on the Galerkin finite element method. The results are discussed based on the governing parameters, such as nanoparticle volume fraction, thermal and solutal Rayleigh numbers. The average Nusselt number was found to increase with the increasing nanoparticle volume fractions. Also, increasing the thermal Rayleigh number enhanced heat transfer while the solutal Rayleigh number has an insignificant effect on it. More importantly, increasing the thermal Rayleigh number assisted avoiding the agglomeration of nanoparticles around the blades and ensured more uniform nanoparticle distribution.

Derivation of Fundamental Solution of Heat Equation by u sing Symmetry Reduction

The objective of this article is to present the fundamental solution of heat equation using symmetry of reduction which is associated with partial derivatives of heat equations through its initial conditions ICs) To emphasize our main results, we also consider some important way of solving of partial differential equation. The main results of our paper are quite general in nature and yield a very large interesting fundamental solution of heat equation and it is used for problems of differential mathematics and mathematical physics special in the area of thermodynamics

Memory-Optimized Wavefront Parallelism on GPUs

Wavefront parallelism is a well-known technique for exploiting the concurrency of applications that execute nested loops with uniform data dependencies. Recent research of such applications, which range from sequence alignment tools to partial differential equation solvers, has used GPUs to benefit from the massively parallel computing resources. To achieve optimal performance, tiling has been introduced as a popular solution to achieve a balanced workload. However, the use of hyperplane tiles increases the cost of synchronization and leads to poor data locality. In this paper, we present a highly optimized implementation of the wavefront parallelism technique that harnesses the GPU architecture. A balanced workload and maximum resource utilization are achieved with an extremely low synchronization overhead. We design the kernel configuration to significantly reduce the minimum number of synchronizations required and also introduce an inter-block lock to minimize the overhead of each synchronization. In addition, shared memory is used in place of the L1 cache. The well-tailored mapping of the operations to the shared memory improves both spatial and temporal locality. We evaluate the performance of our proposed technique for four different applications: sequence alignment, edit distance, summed-area table, and 2D-SOR. The performance results demonstrate that our method achieves speedups of up to six times compared to the previous best-known hyperplane tiling-based GPU implementation.

Chaste: Cancer, Heart and Soft Tissue Environment.

Chaste (Cancer, Heart And Soft Tissue Environment) is an open source simulation package for the numerical solution of mathematical models arising in physiology and biology. To date, Chaste development has been driven primarily by applications that include continuum modelling of cardiac electrophysiology ('Cardiac Chaste'), discrete cell-based modelling of soft tissues ('Cell-based Chaste'), and modelling of ventilation in lungs ('Lung Chaste'). Cardiac Chaste addresses the need for a high-performance, generic, and verified simulation framework for cardiac electrophysiology that is freely available to the scientific community. Cardiac chaste provides a software package capable of realistic heart simulations that is efficient, rigorously tested, and runs on HPC platforms. Cell-based Chaste addresses the need for efficient and verified implementations of cell-based modelling frameworks, providing a set of extensible tools for simulating biological tissues. Computational modelling, along with live imaging techniques, plays an important role in understanding the processes of tissue growth and repair. A wide range of cell-based modelling frameworks have been developed that have each been successfully applied in a range of biological applications. Cell-based Chaste includes implementations of the cellular automaton model, the cellular Potts model, cell-centre models with cell representations as overlapping spheres or Voronoi tessellations, and the vertex model. Lung Chaste addresses the need for a novel, generic and efficient lung modelling software package that is both tested and verified. It aims to couple biophysically-detailed models of airway mechanics with organ-scale ventilation models in a package that is freely available to the scientific community. Chaste is designed to be modular and extensible, providing libraries for common scientific computing infrastructure such as linear algebra operations, finite element meshes, and ordinary and partial differential equation solvers. This infrastructure is used by libraries for specific applications, such as continuum mechanics, cardiac models, and cell-based models. The software engineering techniques used to develop Chaste are intended to ensure code quality, re-usability and reliability. Primary applications of the software include cardiac and respiratory physiology, cancer and developmental biology.

FEM based robust design optimization with Agros and Ārtap

Two powerful and versatile code packages (Agros Suite and Ārtap) are presented for design and optimization of technical devices and systems. Agros Suite represents an environment for numerical solution of systems consisting of partial differential equations (PDEs) of the second order by a higher-order finite element method with a lot of further advanced features such as full adaptivity and selected optimization techniques. Ārtap is a robust design optimization toolbox that provides a simple and efficient programming environment for a wide-range of optimization methods, integrated and external PDE solvers and well-established machine learning tools. Both packages are described in a sufficient detail and their power is illustrated with solution of three problems from the domain of technical sciences.

Redox Flow Batteries: Machine Learning Coupled Multi‐Scale Modeling for Redox Flow Batteries (Adv. Theory Simul. 2/2020)

The framework of a model combining a deep neural network and a partial differential equation solver for redox flow batteries is introduced by Jie Bao, Wei Wang, and co-workers in article number 1900167. Their model establishes a critical link between the micro-structure of a flow-battery component and its performance at the device scale, thereby providing rationale for further operational and material optimization.

Using deep learning to extend the range of air pollution monitoring and forecasting

Across numerous applications, forecasting relies on numerical solvers for partial differential equations (PDEs). Although the use of deep-learning techniques has been proposed, actual applications have been restricted by the fact the training data are obtained using traditional PDE solvers. Thereby, the uses of deep-learning techniques were limited to domains, where the PDE solver was applicable.We demonstrate a deep-learning framework for air-pollution monitoring and forecasting that provides the ability to train across different model domains, as well as a reduction in the run-time by two orders of magnitude. It presents a first-of-a-kind implementation that combines deep-learning and domain-decomposition techniques to allow model deployments extend beyond the domain(s) on which it has been trained.

A 1.87-mm2 56.9-GOPS Accelerator for Solving Partial Differential Equations

Solving partial differential equations (PDEs) require high-precision numerical iterations that are demanding in both computation and memory. We apply the multigrid method with a hybrid layer update to reduce iterations and improve speed, and to transform both fine and coarse grids to a residual form to reduce the precision requirement. The reduced precision enables the mapping of a high-precision PDE solver on SRAMs that perform low-precision parallel multiply-accumulates (MACs) in memory, reducing both energy and area. We employ a delay-locked loop to generate well-controlled unit pulses for driving word lines and a dual-ramp single-slope analog-to-digital converter (ADC) to convert bitline outputs. The design is prototyped in a 1.87-mm2 180-nm test chip made of four 320 × 64 MAC SRAMs, each supporting 128× parallel 5 b × 5 b MACs with 32 5-b output ADCs and consuming 16.6 mW at 200 MHz. The test chip is demonstrated to reach an error tolerance of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-8</sup> in solving PDEs at a grid update rate of 1.38-G entries/s.

M4EKR, Multiphysics for ElectroKinetic Remediation of Polluted Soils

Electrokinetic soil remediation (EKR) is one of the most promising techniques for decontamination of low permeability soils, in which the most classical techniques have been found to be less efficient. However, its practical application on a real scale has been rather limited since the phenomena involved in these processes are very complex. For this reason, it is essential to use numerical models that allow us to know the main trends in the behaviour of soils and natural waters subjected to EKR processes. This study presents the numerical model M4EKR (Multiphysics for ElectroKinetic Remediation). The M4EKR module is a reactive transport model for partially saturated soils that allows reproducing the transport of species due to electroosmosis, electromigration, diffusion and advective flow. The model was completely implemented in COMSOL Multiphysics, a partial differential equation solver. The system of differential and algebraic equations to solve the chemical and transport problem was fully defined by the authors, and they were solved by the M4EKR module in COMSOL (monolithic approach). The scope of the model is illustrated by simulating an EKR process of a natural soil and porewater contaminated with a polar pesticide: 2,4-Dichlorophenoxyacetic acid. For simplicity, the M4EKR version used in this study does not include gas transport, it does not consider the deformability of the soil and it is assumed the processes occur under isothermal conditions.

Nektar++: Enhancing the capability and application of high-fidelity spectral/[formula omitted] element methods

Nektar++ is an open-source framework that provides a flexible, high-performance and scalable platform for the development of solvers for partial differential equations using the high-order spectral/hp element method. In particular, Nektar++ aims to overcome the complex implementation challenges that are often associated with high-order methods, thereby allowing them to be more readily used in a wide range of application areas. In this paper, we present the algorithmic, implementation and application developments associated with our Nektar++ version 5.0 release. We describe some of the key software and performance developments, including our strategies on parallel I/O, on in situ processing, the use of collective operations for exploiting current and emerging hardware, and interfaces to enable multi-solver coupling. Furthermore, we provide details on a newly developed Python interface that enables a more rapid introduction for new users unfamiliar with spectral/hp element methods, C++ and/or Nektar++. This release also incorporates a number of numerical method developments – in particular: the method of moving frames (MMF), which provides an additional approach for the simulation of equations on embedded curvilinear manifolds and domains; a means of handling spatially variable polynomial order; and a novel technique for quasi-3D simulations (which combine a 2D spectral element and 1D Fourier spectral method) to permit spatially-varying perturbations to the geometry in the homogeneous direction. Finally, we demonstrate the new application-level features provided in this release, namely: a facility for generating high-order curvilinear meshes called NekMesh; a novel new AcousticSolver for aeroacoustic problems; our development of a ‘thick’ strip model for the modelling of fluid–structure interaction (FSI) problems in the context of vortex-induced vibrations (VIV). We conclude by commenting on some lessons learned and by discussing some directions for future code development and expansion. Program summaryProgram Title: Nektar++Program Files doi:http://dx.doi.org/10.17632/9drxd9d8nx.1Code Ocean Capsule:https://doi.org/10.24433/CO.9865757.v1Licensing provisions: MITProgramming language: C++External routines/libraries: Boost, METIS, FFTW, MPI, Scotch, PETSc, TinyXML, HDF5, OpenCASCADE, CWIPINature of problem: The Nektar++ framework is designed to enable the discretisation and solution of time-independent or time-dependent partial differential equations.Solution method: spectral/hp element method

Deriving the General Laplace Inversion Formula using Complex Integration Results and its Applications in Solving Partial Differential Equations

Deriving the General Laplace Inversion Formula using Complex Integration Results and its Applications in Solving Partial Differential Equations