scholarly journals A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Prabhat Mishra ◽  
Vikas Gupta ◽  
Ritesh Kumar Dubey
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Mesh Adaptation ◽  
Adaptation Algorithm ◽  
Numerical Tests ◽  
Monitor Function ◽  
Distribution Principle ◽  
Solution Region ◽  
Layer Solution

<p style='text-indent:20px;'>In this work a novel mesh adaptation technique is proposed to approximate discontinuous or boundary layer solution of partial differential equations. We introduce new estimator and monitor function to detect solution region containing discontinuity and layered region. Subsequently, this information is utilized along with equi-distribution principle to adapt the mesh locally. Numerical tests for numerous scalar problems are presented. These results clearly demonstrate the robustness of this method and non-oscillatory nature of the computed solutions.</p>

scholarly journals Solving the Nonlinear Boundary Layer Flow Equations with Pressure Gradient and Radiation

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 710
Author(s):  
Michalis A. Xenos ◽  
Eugenia N. Petropoulou ◽  
Anastasios Siokis ◽  
U. S. Mahabaleshwar
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Pressure Gradient ◽  
Boundary Layer Flow ◽  
Mathematical Formulation ◽  
Dimensionless Temperature ◽  
Partial Differential ◽  
Analytical Results ◽  
Layer Flow

The physical problem under consideration is the boundary layer problem of an incompressible, laminar flow, taking place over a flat plate in the presence of a pressure gradient and radiation. For the mathematical formulation of the problem, the partial differential equations of continuity, energy, and momentum are taken into consideration with the boundary layer simplifications. Using the dimensionless Falkner–Skan transformation, a nonlinear, nonhomogeneous, coupled system of partial differential equations (PDEs) is obtained, which is solved via the homotopy analysis method. The obtained analytical solution describes radiation and pressure gradient effects on the boundary layer flow. These analytical results reveal that the adverse or favorable pressure gradient influences the dimensionless velocity and the dimensionless temperature of the boundary layer. An adverse pressure gradient causes significant changes on the dimensionless wall shear parameter and the dimensionless wall heat-transfer parameter. Thermal radiation influences the thermal boundary layer. The analytical results are in very good agreement with the corresponding numerical ones obtained using a modification of the Keller’s-box method.

scholarly journals Redistribution of Nodes with Two Constraints in Meshless Method of Line to Time-Dependent Partial Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Jafar Biazar ◽  
Mohammad Hosami
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Meshless Method ◽  
Time Dependent ◽  
Arc Length ◽  
Adaptive Meshes ◽  
Partial Differential ◽  
Monitor Function ◽  
Ill Conditioning ◽  
Set Of Points

Meshless method of line is a powerful device to solve time-dependent partial differential equations. In integrating step, choosing a suitable set of points, such as adaptive nodes in spatial domain, can be useful, although in some cases this can cause ill-conditioning. In this paper, to produce smooth adaptive points in each step of the method, two constraints are enforced in Equidistribution algorithm. These constraints lead to two different meshes known as quasi-uniform and locally bounded meshes. These avoid the ill-conditioning in applying radial basis functions. Moreover, to generate more smooth adaptive meshes another modification is investigated, such as using modified arc-length monitor function in Equidistribution algorithm. Influence of them in growing the accuracy is investigated by some numerical examples. The results of consideration of two constraints are compared with each other and also with uniform meshes.

scholarly journals A note on the non-inertial similarity solution for von Kármán swirling flow

2020 ◽  
pp. 2150030
Author(s):  
Madeleine L. Combrinck
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Similarity Solution ◽  
Shooting Method ◽  
Swirling Flow ◽  
Boundary Layer Equations ◽  
Von Karman ◽  
Von Kármán

This note proposes a non-inertial similarity solution for the classic von Kármán swirling flow as perceived from the rotational frame. The solution is obtained by implementing non-inertial similarity parameters in the non-inertial boundary layer equations. This reduces the partial differential equations to a set of ordinary differential equations that is solved through an integration routine and shooting method.

scholarly journals On the Multidomain Bivariate Spectral Local Linearisation Method for Solving Systems of Nonsimilar Boundary Layer Partial Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Vusi Mpendulo Magagula
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Large Time ◽  
Nonlinear Partial Differential Equations ◽  
Computation Time ◽  
General System ◽  
Partial Differential ◽  
Boundary Layer Equations ◽  
Grid Points

In this work, a novel approach for solving systems of nonsimilar boundary layer equations over a large time domain is presented. The method is a multidomain bivariate spectral local linearisation method (MD-BSLLM), Legendre-Gauss-Lobatto grid points, a local linearisation technique, and the spectral collocation method to approximate functions defined as bivariate Lagrange interpolation. The method is developed for a general system of n nonlinear partial differential equations. The use of the MD-BSLLM is demonstrated by solving a system of nonlinear partial differential equations that describe a class of nonsimilar boundary layer equations. Numerical experiments are conducted to show applicability and accuracy of the method. Grid independence tests establish the accuracy, convergence, and validity of the method. The solution for the limiting case is used to validate the accuracy of the MD-BSLLM. The proposed numerical method performs better than some existing numerical methods for solving a class of nonsimilar boundary layer equations over large time domains since it converges faster and uses few grid points to achieve accurate results. The proposed method uses minimal computation time and its accuracy does not deteriorate with an increase in time.

A Subdivision-based Geometric Multigrid Method

2021 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Multigrid Method ◽  
Benchmark Problems ◽  
Complex Geometries ◽  
Numerical Tests ◽  
Surface Subdivision ◽  
Regularizing Effects ◽  
Geometric Multigrid Method ◽  
Geometric Multigrid

In this paper we introduce a smooth subdi- vision theory-based geometric multigrid method. While theory and efficiency of geometric multigrid methods rely on grid regularity, this requirement is often not directly fulfilled in applications where partial differential equations are defined on complex geometries. Instead of generating multigrid hierarchies with classical linear refinement, we here propose the use of smooth subdivision theory for automatic grid hierarchy regularization within a geometric multigrid solver. This subdivi- sion multigrid method is compared to the classical geometric multigrid method for two benchmark problems. Numerical tests show significant improvement factors for iteration numbers and solve times when comparing subdivision to classical multigrid. A second study fo- cusses on the regularizing effects of surface subdivision refinement, using the Poisson-Nernst-Planck equations. Subdivision multigrid is demonstrated to outperform classical multigrid.

Solution methodology of convection boundary layer problems using Adomian decomposition method

2012 ◽  
Vol 227 (7) ◽  
pp. 1554-1565
Author(s):  
MJ Javanmardi ◽  
K Jafarpur ◽  
M Mahzoon
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Convection Boundary Layer ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
Convection Boundary ◽  
Boundary Layer Problems

Adomian decomposition method has been applied to some convection boundary layer problems. It is shown that the method is excellent in solving nonlinear partial differential equations. Comparison of the results so obtained and those from conventional methods proves the technique to be powerful and to give nearly exact solutions. Since most physical problems are governed by nonlinear ordinary or partial differential equations, application of this method may be advised to simplify the method of solution.

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