Localized method of approximate particular solutions for solving unsteady Navier–Stokes problem

Abstract

Highlights• The LMAPS is proposed to solve Navier-Stokes equations.• The weighting coefficients are obtained by local MQ-RBFs.• A global and sparse matrix is reformulated.• Fractional step algorithm is adopted. AbstractThe localized method of approximate particular solutions (LMAPS) is proposed to solve two-dimensional transient incompressible Navier-Stokes systems of equations in primitive variables. The equations contain the Laplacian operator. In avoiding ill-conditioning problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear system within local supporting domain in which five nearest neighboring points and multiquadrics are used for interpolation. Then local matrices are reformulated in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The method is assessed on driven cavity problem and flow around cylinder. The numerical experiments show that the newly developed LMAPS is suitable for solving incompressible Navier-Stokes equations with high accuracy and efficiency.

Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials

“…To the best of our current knowledge, the meshless schemes, including Kansa method [14], method of fundamental solutions [15], method of particular solutions [16,17], element-free Galerkin method [18], local point interpolation [19], and boundary knot method [20], are widely used to approximate a large class of partial differential equations in science and engineering fields. As reported in the literatures, the MPS has been applied to solve the Navier-Stokes problem [21], wave propagation problem [22], and time-fractional diffusion problem [23]. Despite the effectiveness of the MPS, there are some disadvantages such as the ill-conditioned collocation matrix, the uncertainty of the shape parameters, and difficulties in deriving the closed-form particular solutions for general differential operators, and for more details, please refer to [12,17,[24][25][26] and the references cited therein.…”

Colocación con funciones de base radial para la solución de modelos matemáticos en fenoménos de transporte

“…Sin embargo, en la implementación de las formulaciones locales, se pierde parte del carácter sin malla del esquema numérico, debido a la necesidad de definir la conectividad local parcial de los nodos de interpolación como se propone en los trabajos de Stevens y Power (Stevens, Power, Meng, Howard, y Cliffe, 2013). Las formulaciones locales han resultado ser mas eficientes en muchos problemas, por ejemplo en la predicción de campos de flujo viscoso a un número de Reynolds moderado y alto (Zhang, Chen, Chen, y Li, 2016), y otros casos no lineales como flujo a través de medios porosos en suelos (Jackson, Power, Giddings, y Stevens, 2017). La solución numérica por RBF de ecuaciones diferenciales parciales también puede mejorarse mediante métodos simétricos hermíticos, que interpolan las variables en los nodos y además aplica y cumple exactamente el operador diferencial en un conjunto de puntos.…”

Numerical Simulation of Non-Linear Schrödinger Equations in Arbitrary Domain by the Localized Method of Approximate Particular Solution

“…In the LMAPS, the k-d tree algorithm [38] is applied for searching the neighboring points of each local center. Recently, the LMAPS has been successfully applied for the simulation of heat conduction, the molecular dynamics, the wave propagation, and the flow fields [39][40][41][42].…”