A numerical solution algorithm for the spatially inhomogeneous equations of kinetic breakdown

1988 ◽  
Vol 55 (4) ◽  
pp. 1154-1160 ◽  
Author(s):  
M. V. Bochkov ◽  
B. N. Chetverushkin
Keyword(s):  
Numerical Solution ◽  
Solution Algorithm ◽  
Spatially Inhomogeneous

scholarly journals Numerical Solution and Simulation of Second-Order Parabolic PDEs with Sinc-Galerkin Method Using Maple

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Aydin Secer
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Equation System ◽  
Solution Algorithm ◽  
Basis Functions ◽  
Algebraic Equations ◽  
Parabolic Pdes ◽  
Linear Partial Differential Equations ◽  
Type Boundary

An efficient solution algorithm for sinc-Galerkin method has been presented for obtaining numerical solution of PDEs with Dirichlet-type boundary conditions by using Maple Computer Algebra System. The method is based on Whittaker cardinal function and uses approximating basis functions and their appropriate derivatives. In this work, PDEs have been converted to algebraic equation systems with new accurate explicit approximations of inner products without the need to calculate any numeric integrals. The solution of this system of algebraic equations has been reduced to the solution of a matrix equation system via Maple. The accuracy of the solutions has been compared with the exact solutions of the test problem. Computational results indicate that the technique presented in this study is valid for linear partial differential equations with various types of boundary conditions.

scholarly journals Numerical Modeling of Self-Consistent Dynamics of Shallow and Ground Waters

2021 ◽  
pp. 45-62
Author(s):  
Sergey Khrapov
Keyword(s):  
Numerical Modeling ◽  
Numerical Solution ◽  
Software Package ◽  
Simulation Software ◽  
Second Order ◽  
Order Of Accuracy ◽  
Ground Waters ◽  
Spatially Inhomogeneous ◽  
Tvd Method ◽  
Joint Dynamics

A mathematical and numerical model of the joint dynamics of shallow and ground waters has been built, which takes into account the nonlinear dynamics of a liquid, water absorption from the surface into the ground, filtration currents in the ground, and water seepage from the ground back to the surface. The dynamics of shallow waters is described by the Saint-Venant equations, taking into account the spatially inhomogeneous distributions of the terrain, the coefficients of bottom friction and infiltration, as well as non-stationary sources and flows of water. For the numerical integration of Saint-Venant’s equations, the well-tested CSPH-TVD method of the second order of accuracy is used, the parallel CUDA algorithm of which is implemented as a software package “EcoGIS-Simulation” for high-performance computing on supercomputers with graphic coprocessors (GPU). The dynamics of groundwater is described by the nonlinear Bussensk equation, generalized to the case of a spatially inhomogeneous distribution of the parameters of the porous medium and the surface of the aquiclude (the boundary between water-permeable and low-permeable soils). The numerical solution of this equation is built on the basis of a finite-difference scheme of the second order of accuracy, the CUDA algorithm of which is integrated into the calculation module of the “EcoGIS-Simulation” software package and is consistent with the main stages of the CSPH-TVD method. The relative deviation of the numerical solution from the exact solution of the nonlinear Boussinesq equation does not exceed 10−4–10−5. The paper compares the results of numerical modeling of the dynamics of groundwater with analytical solutions of the linearized Bussensk equation used as calculation formulas in the methods for predicting the level of groundwater in the vicinity of water bodies. It is shown that the error of these methods is several percent even for the simplest case of a plane-parallel flow of groundwater with a constant backwater. Based on the results obtained, it was concluded that the proposed method for numerical modeling of the joint dynamics of surface and ground waters can be more versatile and efficient (it has significantly better accuracy and productivity) in comparison with the existing methods for calculating flooding zones, especially for hydrodynamic flows with complex geometry and nonlinear interaction of counter fluid flows arising during seasonal floods during flooding of vast land areas.

scholarly journals SOLA-DM: A numerical solution algorithm for transient three-dimensional flows

1988 ◽  
Author(s):  
T.L. Wilson ◽  
B.D. Nichols ◽  
C.W. Hirt ◽  
L.R. Stein
Keyword(s):  
Numerical Solution ◽  
Three Dimensional ◽  
Solution Algorithm
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