scholarly journals Applying the Swept Rule for Solving Two-Dimensional Partial Differential Equations on Heterogeneous Architectures

2021 ◽  
Vol 26 (3) ◽  
pp. 52
Author(s):  
Anthony S. Walker ◽  
Kyle E. Niemeyer
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Heterogeneous Computing ◽  
Heat Diffusion ◽  
Fixed Cost ◽  
Two Dimensional ◽  
Partial Differential ◽  
Computing Systems ◽  
Heat Diffusion Equation ◽  
Work Distribution

The partial differential equations describing compressible fluid flows can be notoriously difficult to resolve on a pragmatic scale and often require the use of high-performance computing systems and/or accelerators. However, these systems face scaling issues such as latency, the fixed cost of communicating information between devices in the system. The swept rule is a technique designed to minimize these costs by obtaining a solution to unsteady equations at as many possible spatial locations and times prior to communicating. In this study, we implemented and tested the swept rule for solving two-dimensional problems on heterogeneous computing systems across two distinct systems and three key parameters: problem size, GPU block size, and work distribution. Our solver showed a speedup range of 0.22–2.69 for the heat diffusion equation and 0.52–1.46 for the compressible Euler equations. We can conclude from this study that the swept rule offers both potential for speedups and slowdowns and that care should be taken when designing such a solver to maximize benefits. These results can help make decisions to maximize these benefits and inform designs.

Advanced Image Analysis for Learning Underlying Partial Differential Equations for Anomaly Identification

2020 ◽  
Vol 64 (2) ◽  
pp. 20510-1-20510-10
Author(s):  
Andrew Miller ◽  
Jan Petrich ◽  
Shashi Phoha
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Time ◽  
Heat Diffusion ◽  
Series Data ◽  
Model Parameters ◽  
Model Structure ◽  
Partial Differential ◽  
Individual Parameter ◽  
Heat Diffusion Equation

Abstract In this article, the authors adapt and utilize data-driven advanced image processing and machine learning techniques to identify the underlying dynamics and the model parameters for dynamic processes driven by partial differential equations (PDEs). Potential applications include non-destructive inspection for material crack detection using thermal imaging as well as real-time anomaly detection for process monitoring of three-dimensional printing applications. A neural network (NN) architecture is established that offers sufficient flexibility for spatial and temporal derivatives to capture the physical dependencies inherent in the process. Predictive capabilities are then established by propagating the process forward in time using the acquired model structure as well as individual parameter values. Moreover, deviations in the predicted values can be monitored in real time to detect potential process anomalies or perturbations. For concept development and validation, this article utilizes well-understood PDEs such as the homogeneous heat diffusion equation. Time series data governed by the heat equation representing a parabolic PDE is generated using high-fidelity simulations in order to construct the heat profile. Model structure and parameter identification are realized through a shallow residual convolutional NN. The learned model structure and associated parameters resemble a spatial convolution filter, which can be applied to the current heat profile to predict the diffusion behavior forward in time.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2620-2649 ◽  
Author(s):  
Rajni Rohila ◽  
R.C. Mittal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Spline Functions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

scholarly journals SOLVING SYSTEMS OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA TWO DIMENSIONAL LAPLACE TRANSFORMS

2020 ◽  
Vol 5 (12) ◽  
pp. 406-420
Author(s):  
A. Aghili ◽  
M.R. Masomi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Laplace Transforms ◽  
Heat Equations ◽  
Two Dimensional ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Homogeneous Systems ◽  
Fractional Pde

In this article, the authors used two dimensional Laplace transform to solve non - homogeneous sub - ballistic fractional PDE and homogeneous systems of time fractional heat equations. Constructive examples are also provided.

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

2013 ◽  
Vol 5 (2) ◽  
pp. 212-221
Author(s):  
Houguo Li ◽  
Kefu Huang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Partial Differential ◽  
Isotropic Materials ◽  
Group Theoretical Method ◽  
Infinitesimal Operators

AbstractInvariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoretical method. The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators. Three invariant solutions are constructed. Their graphical figures are presented and physical meanings are elucidated in some cases.

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