# A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations

## Abstract

Many difficult problems arise in the numerical simulation of fluid flow processes within porous media in petroleum reservoir simulation and in subsurface contaminant transport and remediation. The authors develop a family of Eulerian-Lagrangian localized adjoint methods for the solution of the initial-boundary value problems for first-order advection-reaction equations on general multi-dimensional domains. Different tracking algorithms, including the Euler and Runge-Kutta algorithms, are used. The derived schemes, which are full mass conservative, naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary conditions. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices, which can be efficiently solved by the conjugate gradient method in an optimal order number of iterations without any preconditioning needed. Numerical results are presented to compare the performance of the ELLAM schemes with many well studied and widely used methods, including the upwind finite difference method, the Galerkin and the Petrov-Galerkin finite element methods with backward-Euler or Crank-Nicolson temporal discretization, the streamline diffusion finite element methods, the monotonic upstream-centered scheme for conservation laws (MUSCL), and the Minmod scheme.

- Authors:

- Univ. of South Carolina, Columbia, SC (United States). Dept. of Mathematics
- Texas A and M Univ., College Station, TX (United States). Inst for Scientific Computation
- Mobil Technology Co., Dallas, TX (United States). Upstream Strategic Research Center
- Sultan Qaboos Univ., Muscat (Oman). Dept. of Mathematics and Statistics

- Publication Date:

- OSTI Identifier:
- 361771

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 152; Journal Issue: 1; Other Information: PBD: 10 Jun 1999

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 54 ENVIRONMENTAL SCIENCES; 02 PETROLEUM; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; RESERVOIR ENGINEERING; OIL FIELDS; ENVIRONMENTAL TRANSPORT; POLLUTANTS; ADJOINT DIFFERENCE METHOD; BOUNDARY-VALUE PROBLEMS; ADVECTION; ALGORITHMS; NUMERICAL SOLUTION

### Citation Formats

```
Wang, H., Man, S., Ewing, R.E., Qin, G., Lyons, S.L., and Al-Lawatia, M.
```*A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations*. United States: N. p., 1999.
Web. doi:10.1006/jcph.1999.6239.

```
Wang, H., Man, S., Ewing, R.E., Qin, G., Lyons, S.L., & Al-Lawatia, M.
```*A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations*. United States. doi:10.1006/jcph.1999.6239.

```
Wang, H., Man, S., Ewing, R.E., Qin, G., Lyons, S.L., and Al-Lawatia, M. Thu .
"A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations". United States. doi:10.1006/jcph.1999.6239.
```

```
@article{osti_361771,
```

title = {A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations},

author = {Wang, H. and Man, S. and Ewing, R.E. and Qin, G. and Lyons, S.L. and Al-Lawatia, M.},

abstractNote = {Many difficult problems arise in the numerical simulation of fluid flow processes within porous media in petroleum reservoir simulation and in subsurface contaminant transport and remediation. The authors develop a family of Eulerian-Lagrangian localized adjoint methods for the solution of the initial-boundary value problems for first-order advection-reaction equations on general multi-dimensional domains. Different tracking algorithms, including the Euler and Runge-Kutta algorithms, are used. The derived schemes, which are full mass conservative, naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary conditions. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices, which can be efficiently solved by the conjugate gradient method in an optimal order number of iterations without any preconditioning needed. Numerical results are presented to compare the performance of the ELLAM schemes with many well studied and widely used methods, including the upwind finite difference method, the Galerkin and the Petrov-Galerkin finite element methods with backward-Euler or Crank-Nicolson temporal discretization, the streamline diffusion finite element methods, the monotonic upstream-centered scheme for conservation laws (MUSCL), and the Minmod scheme.},

doi = {10.1006/jcph.1999.6239},

journal = {Journal of Computational Physics},

number = 1,

volume = 152,

place = {United States},

year = {1999},

month = {6}

}