ERROR ESTIMATE FOR A SPLITTING METHOD APPLIED TO CONVECTION-REACTION EQUATIONS

2001 ◽  
Vol 11 (06) ◽  
pp. 1081-1100 ◽  
Author(s):  
F. PEYROUTET ◽  
M. MADAUNE-TORT
Keyword(s):  
Error Estimate ◽  
Homogeneous Problem ◽  
Operator Splitting ◽  
Splitting Method ◽  
Initial Boundary ◽  
Approximate Solutions ◽  
Initial Boundary Value Problems ◽  
Operator Splitting Method ◽  
Initial Boundary Value ◽  
Reaction Equations

An operator splitting method is used to approximate solutions of initial–boundary value problems related to a hyperbolic reaction equation. In this paper, we study the order of convergence for this scheme using only bounded entropy weak solutions of the associated homogeneous problem. An L1-error estimate of order 1 is proved by means of Kruzkov's comparison techniques.

scholarly journals Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

Geosciences ◽  
2019 ◽  
Vol 9 (1) ◽  
pp. 29 ◽  
Author(s):  
Paul M. Delgado ◽  
V. M. Krushnarao Kotteda ◽  
Vinod Kumar
Keyword(s):  
Fixed Point ◽  
Asymptotic Solution ◽  
Relative Error ◽  
Operator Splitting ◽  
Splitting Method ◽  
Approximate Solutions ◽  
Time Step ◽  
Operator Splitting Method ◽  
The Stability ◽  
Stability And Consistency

Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 − 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 − 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 − 6 .

Recent Advances and Applications of Reduction Methods

1994 ◽  
Vol 47 (5) ◽  
pp. 125-146 ◽  
Author(s):  
Ahmed K. Noor
Keyword(s):  
Eigenvalue Problems ◽  
Operator Splitting ◽  
Nonlinear Dynamic Analysis ◽  
Analytical Techniques ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Future Directions ◽  
Reduction Techniques ◽  
Reduction Methods ◽  
Initial Boundary Value

The focus of this review is on the basic idea and useful interpretations of reduction methods; multiple-parameter reduction methods; treatment of nonconservative loadings (unsymmetric systems); and application of reduction methods in conjunction with operator splitting. The literature reviewed is devoted to the mathematical aspects of reduction methods, as well as to the following seven application areas: eigenvalue problems; nonlinear vibrations; linear and nonlinear dynamic analysis (initial/boundary value problems); linear systems analyzed by using semi-analytic numerical discretization procedures; reanalysis techniques; sensitivity analysis; and optimum design. Sample numerical results are presented showing the effectiveness of reduction methods in a recent application. Hybrid analytical techniques which share some of the key elements with reduction techniques are highlighted. Some of the future directions for research on reduction methods are outlined.

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