SECOND-ORDER METHOD FOR SOLVING 2D NONLINEAR PARABOLIC DIFFERENTIAL EQUATIONS BASED ON ADI METHOD

2010 ◽  
Vol 01 (01) ◽  
pp. 133-146 ◽  
Author(s):  
SADEGH AMIRI ◽  
S. MOHAMMAD HOSSEINI
Keyword(s):  
Parabolic Problems ◽  
Computationally Efficient ◽  
Parabolic Differential Equations ◽  
Adi Method ◽  
Tridiagonal Matrices ◽  
Nonlinear Parabolic ◽  
Alternating Direction ◽  
Method Of Solution ◽  
Diagonally Dominant ◽  
Bounded Sets

In this article, a new unconditionally stable method based on alternating direction implicit (ADI) method for solving nonlinear unsteady 2D parabolic problems is presented. The order of this method in both time and space is 2. For this development, the Peaceman–Rachford ADI method has been modified suitably to take care of nonlinear forcing term of the equation appropriately. The unconditional stability of the method is shown under discrete L2 norm on bounded sets. The method of solution consists of a number of strictly diagonally dominant tridiagonal matrices, which make the method computationally efficient. The accuracy of the proposed method is also demonstrated by some numerical examples.

scholarly journals Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms

2013 ◽  
Vol 143 (6) ◽  
pp. 1185-1208 ◽  
Author(s):  
Rosaria Di Nardo ◽  
Filomena Feo ◽  
Olivier Guibé
Keyword(s):  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Growth Conditions ◽  
Dirichlet Boundary Conditions ◽  
Parabolic Problems ◽  
Renormalized Solutions ◽  
Uniqueness Results ◽  
Nonlinear Parabolic ◽  
Image Position ◽  
Lower Order Terms

We consider a general class of parabolic equations of the typewith Dirichlet boundary conditions and with a right-hand side belonging to L1 + Lp′ (W−1, p′). Using the framework of renormalized solutions we prove uniqueness results under appropriate growth conditions and Lipschitz-type conditions on a(u, ∇u), K(u) and H(∇u).

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