Mass-preserving time second-order explicit–implicit domain decomposition schemes for solving parabolic equations with variable coefficients

2018 ◽  
Vol 37 (4) ◽  
pp. 4423-4442 ◽  
Author(s):  
Zhongguo Zhou ◽  
Dong Liang
Keyword(s):  
Domain Decomposition ◽  
Parabolic Equations ◽  
Variable Coefficients ◽  
Second Order ◽  
Decomposition Schemes

LAGRANGIAN FEYNMAN FORMULAS FOR SECOND-ORDER PARABOLIC EQUATIONS IN BOUNDED AND UNBOUNDED DOMAINS

2010 ◽  
Vol 13 (03) ◽  
pp. 377-392 ◽  
Author(s):  
YANA A. BUTKO ◽  
MARTIN GROTHAUS ◽  
OLEG G. SMOLYANOV
Keyword(s):  
Parabolic Equations ◽  
Diffusion Processes ◽  
Unbounded Domains ◽  
Variable Coefficients ◽  
Second Order ◽  
Cauchy Problems ◽  
Elementary Functions ◽  
Functional Integrals ◽  
Finite Dimensional ◽  
Variable Diffusion

In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.

scholarly journals A Substructuring Domain Decomposition Scheme for Unsteady Problems

2011 ◽  
Vol 11 (2) ◽  
pp. 241-268 ◽  
Author(s):  
Petr Vabishchevich
Keyword(s):  
Domain Decomposition ◽  
Partition Of Unity ◽  
Decomposition Methods ◽  
Second Order ◽  
Computing Systems ◽  
Decomposition Scheme ◽  
Initial Domain ◽  
Unsteady Problems ◽  
Decomposition Schemes ◽  
Free Domain

AbstractDomain decomposition methods are used for the approximate solution of boundary-value problems for partial differential equations on parallel computing systems. Specific features of unsteady problems are fully taken into account in iteration-free domain decomposition schemes. Regionally-additive schemes are based on various classes of splitting schemes. In this paper we highlight a class of domain decomposition schemes which are based on the partition of the initial domain into subdomains with common boundary nodes. Using a partition of unity we construct and analyze unconditionally stable schemes for domain decomposition based on a two-component splitting: the problem within each subdomain and the problem at their boundaries. As an example we consider a Cauchy problem of first or second order in time with a non-negative self-adjoint second order operator in space. The theoretical discussion is supplemented with the numerical solution of a model problem for a two-dimensional parabolic equation.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

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