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==Abstract== | ==Abstract== | ||
| − | We present the design of a high-resolution Petrov–Galerkin (HRPG) method using linear finite elements | + | We present the design of a high-resolution Petrov–Galerkin (HRPG) method using linear finite elements for the problem defined by the residual <math>R (\phi):= \partial \phi /\partial t + u \partial \phi /\partial x - k \partial^2 \phi /\partial x^2 +s \phi -f</math> where <math>k; \ge s 0</math>. The structure of the method in 1D is identical to the consistent approximate upwind Petrov– |
| − | for the problem defined by the residual | + | Galerkin (CAU/PG) method [A.C. Galeão, E.G. Dutra do Carmo, A consistent approximate upwind Petrov–Galerkin method for the convection-dominated problems, Comput. Methods Appl. Mech. Engrg. 68(1988) 83–95] except for the definitions of the stabilization parameters. Such a structure may also be attained via the finite-calculus (FIC) procedure [E. Oñate, Derivation of stabilized equations for numerical solution of advective–diffusive transport and fluid flow problems, Comput. Methods Appl. Mech. Engrg. 151 (1998) 233–265; E. Oñate, J. Miquel, G. Hauke, Stabilized formulation for the advection–diffusion– |
| − | + | absorption equation using finite-calculus and linear finite elements, Comput. Methods Appl. Mech. Engrg. 195 (2006) 3926–3946] by an appropriate definition of the characteristic length. The prefix ‘high-resolution’ is used here in the sense popularized by Harten, i.e. second order accuracy for smooth/regular regimes | |
| − | <math>R (\phi):= \partial \phi /\partial t + u \partial \phi /\partial x - k \partial^2 \phi /\partial x^2 +s \phi -f</math> | + | and good shock-capturing in nonregular regimes. The design procedure embarks on the problem of circumventing the Gibbs phenomenon observed in L2-projections. Next we study the conditions on |
| − | + | the stabilization parameters to circumvent the global oscillations due to the convective term. A conjuncture of the two results is made to deal with the problem at hand that is usually plagued by Gibbs, global | |
| − | where <math>k; \ge s 0</math>. The structure of the method in 1D is identical to the consistent approximate upwind Petrov– | + | and dispersive oscillations in the numerical solution. It is shown that the method indeed reproduces stabilized high-resolution numerical solutions for a wide range of values of <math>u; k; s</math> and <math>f</math>. Finally, some remarks are made on the extension of the HRPG method to multidimensions. |
| − | Galerkin (CAU/PG) method [A.C. Galeão, E.G. Dutra do Carmo, A consistent approximate upwind | + | |
| − | + | ||
| − | (1988) 83–95] except for the definitions of the stabilization parameters. Such a structure may also be attained | + | |
| − | via the finite-calculus (FIC) procedure [E. Oñate, Derivation of stabilized equations for numerical | + | |
| − | solution of advective–diffusive transport and fluid flow problems, Comput. Methods Appl. Mech. Engrg. | + | |
| − | 151 (1998) 233–265; E. Oñate, J. Miquel, G. Hauke, Stabilized formulation for the advection–diffusion– | + | |
| − | absorption equation using finite-calculus and linear finite elements, Comput. Methods Appl. Mech. Engrg. | + | |
| − | 195 (2006) 3926–3946] by an appropriate definition of the characteristic length. The prefix ‘high-resolution’ | + | |
| − | is used here in the sense popularized by Harten, i.e. second order accuracy for smooth/regular regimes | + | |
| − | and good shock-capturing in nonregular regimes. The design procedure embarks on the problem | + | |
| − | of circumventing the Gibbs phenomenon observed in L2-projections. Next we study the conditions on | + | |
| − | the stabilization parameters to circumvent the global oscillations due to the convective term. A conjuncture | + | |
| − | of the two results is made to deal with the problem at hand that is usually plagued by Gibbs, global | + | |
| − | and dispersive oscillations in the numerical solution. It is shown that the method indeed reproduces stabilized | + | |
| − | high-resolution numerical solutions for a wide range of values of <math>u; k; s</math> and <math>f</math>. Finally, some remarks | + | |
| − | are made on the extension of the HRPG method to multidimensions. | + | |
| + | ==Full document== | ||
| − | + | https://www.sciencedirect.com/science/article/abs/pii/S0045782509003545 | |
Latest revision as of 12:21, 7 November 2024
Abstract
We present the design of a high-resolution Petrov–Galerkin (HRPG) method using linear finite elements for the problem defined by the residual where . The structure of the method in 1D is identical to the consistent approximate upwind Petrov– Galerkin (CAU/PG) method [A.C. Galeão, E.G. Dutra do Carmo, A consistent approximate upwind Petrov–Galerkin method for the convection-dominated problems, Comput. Methods Appl. Mech. Engrg. 68(1988) 83–95] except for the definitions of the stabilization parameters. Such a structure may also be attained via the finite-calculus (FIC) procedure [E. Oñate, Derivation of stabilized equations for numerical solution of advective–diffusive transport and fluid flow problems, Comput. Methods Appl. Mech. Engrg. 151 (1998) 233–265; E. Oñate, J. Miquel, G. Hauke, Stabilized formulation for the advection–diffusion– absorption equation using finite-calculus and linear finite elements, Comput. Methods Appl. Mech. Engrg. 195 (2006) 3926–3946] by an appropriate definition of the characteristic length. The prefix ‘high-resolution’ is used here in the sense popularized by Harten, i.e. second order accuracy for smooth/regular regimes and good shock-capturing in nonregular regimes. The design procedure embarks on the problem of circumventing the Gibbs phenomenon observed in L2-projections. Next we study the conditions on the stabilization parameters to circumvent the global oscillations due to the convective term. A conjuncture of the two results is made to deal with the problem at hand that is usually plagued by Gibbs, global and dispersive oscillations in the numerical solution. It is shown that the method indeed reproduces stabilized high-resolution numerical solutions for a wide range of values of and . Finally, some remarks are made on the extension of the HRPG method to multidimensions.
Full document
https://www.sciencedirect.com/science/article/abs/pii/S0045782509003545
Document information
Published on 01/01/2010
DOI: 10.1016/j.cma.2009.10.009
Licence: CC BY-NC-SA license
