scholarly journals A Meshless Collocation Method with Barycentric Lagrange Interpolation for Solving the Helmholtz Equation

2021 ◽  
Vol 126 (1) ◽  
pp. 25-54
Author(s):  
Miaomiao Yang ◽  
WentaoMa ◽  
Yongbin Ge
Keyword(s):  
Helmholtz Equation ◽  
Collocation Method ◽  
Lagrange Interpolation ◽  
Meshless Collocation

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Miaomiao Yang ◽  
Xinkun Du ◽  
Yongbin Ge
Keyword(s):  
Differential Equations ◽  
Helmholtz Equation ◽  
Collocation Method ◽  
Meshless Method ◽  
Lagrange Interpolation ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Wave Number ◽  
Content Type ◽  
Meshless Collocation

PurposeThis meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.Design/methodology/approachIn this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.FindingsThe numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.Research limitations/implicationsCompared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.Practical implicationsThe Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.Originality/valueThis meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

Upwinding Meshfree Point Collocation Method for Unsteady Magnetohydrodynamic Flow at High Hartmann Numbers

2011 ◽  
Vol 130-134 ◽  
pp. 1668-1671
Author(s):  
Xing Hui Cai ◽  
Cheng Ying Shi ◽  
Guo Liang Wang
Keyword(s):  
Collocation Method ◽  
Numerical Solutions ◽  
Hartmann Number ◽  
Mhd Flow ◽  
Rectangular Section ◽  
Point Collocation ◽  
Coupled Equations ◽  
Meshless Collocation ◽  
Support Domain ◽  
Flow Through

In this paper, a meshfree point collocation method, with an upwinding scheme, is presented to obtain the numerical solutions of the coupled equations in the velocity field for the unsteady magnetohydrodynamic (MHD) flow through a straight duct of rectangular section with insulated walls. Computations have been carried out for the unsteady MHD flow, which is under the external applied magnetic field of arbitrary orientation, of different Hartmann number from 5 to 106 and at various time levels. As the adaptive upwinding local support domain is introduced in the meshless collocation method, numerical results show that the method can compute MHD problems with Hartmann numbers up to 106 with good accuracy. The results also show that as Hartmann number increases, the time needed to reach the steady state decreases.

Upwinding Meshfree Point Collocation Method for Steady MHD Flow

2010 ◽  
Author(s):  
Xinghui Cai ◽  
Guanghui Su ◽  
Suizheng Qiu
Keyword(s):  
Magnetic Field ◽  
Collocation Method ◽  
Mhd Flow ◽  
Least Square ◽  
Good Convergence ◽  
Moving Least Square ◽  
Point Collocation ◽  
Coupled Equations ◽  
Meshless Collocation ◽  
Mls Approximation

In this paper, a meshfree point collocation method, with a upwinding scheme, is presented to obtain the numerical solution of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic (MHD) flow through a straight pipe of rectangular section with insulated walls. The moving least-square (MLS) approximation is employed to construct shape functions in conjunction with the framework of point collocation method. Computations have been carried out for different applied magnetic field orientations and different Hartmann numbers from 5 to 1,000,000. As the adaptive upwinding local support domain is introduced in the meshless collocation method, numerical results show that the method can compute MHD problems not only at low and moderate values but also at high values of the Hartmann number with high accuracy and good convergence.

Meshless Collocation Method for Inverse Source Identification Problems

2015 ◽  
Vol 7 (4) ◽  
pp. 496-509 ◽  
Author(s):  
Fuzhang Wang ◽  
Zhaoxing Ma
Keyword(s):  
Collocation Method ◽  
Radial Basis Functions ◽  
Source Identification ◽  
Basis Functions ◽  
Computationally Efficient ◽  
Identification Problems ◽  
Present Scheme ◽  
Numerical Examples ◽  
Meshless Collocation ◽  
Boundary Measurements

AbstractA novel meshless scheme is proposed for inverse source identification problems of Helmholtz-type equations. It is formulated by the non-singular general solutions of the Helmholtz-type equations augmented with radial basis functions. Under this meshless scheme, we can determine smooth source terms from partially accessible boundary measurements with accurate results. Numerical examples are presented to verify validity and accuracy of the present scheme. It is demonstrated that the present scheme is simple, accurate, stable and computationally efficient for inverse smooth source identification problems.

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