SYMMETRY BREAKING IN CFD: CAUSES AND CONSEQUENCES

1994 ◽  
Vol 05 (02) ◽  
pp. 189-194 ◽  
Author(s):  
KARL GUSTAFSON ◽  
JOHN McARTHUR
Keyword(s):  
Fluid Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Symmetry Breaking ◽  
Steady Flow ◽  
Time Dependent ◽  
Partial Differential ◽  
One Dimensional ◽  
Computational Schemes ◽  
Three Space

Symmetry breaking occurs in the discretizations of the partial differential equations of fluid dynamics, both advertently and inadvertently. Although it can occur even in one-dimensional steady flow algorithms, we have found its consequences to be more pronounced in two and three space dimensions and in the computation of time dependent flows. This has led us to some interesting new computational schemes.

scholarly journals The decomposition method for linear, one-dimensional, time-dependent partial differential equations

2006 ◽  
Vol 2006 ◽  
pp. 1-29 ◽  
Author(s):  
D. Lesnic
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Mathematical Physics ◽  
Time Dependent ◽  
Partial Differential ◽  
One Dimensional ◽  
Lateral Boundary Conditions ◽  
Adomian Decomposition ◽  
Equations Of Mathematical Physics

The analytical solutions for linear, one-dimensional, time-dependent partial differential equations subject to initial or lateral boundary conditions are reviewed and obtained in the form of convergent Adomian decomposition power series with easily computable components. The efficiency and power of the technique are shown for wide classes of equations of mathematical physics.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

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