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Numerical Solution Of Reservoir Flow Models Based On Large Time Step Operator Splitting Algorithms
 FILTRATION IN POROUS MEDIA AND INDUSTRIAL APPLICATIONS, LECTURE NOTES IN MATHEMATICS
, 1999
"... During recent years the authors and collaborators have been involved in an activity related to the construction and analysis of large time step operator splitting algorithms for the numerical simulation of multiphase flow in heterogeneous porous media. The purpose of these lecture notes is to revie ..."
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Cited by 25 (14 self)
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During recent years the authors and collaborators have been involved in an activity related to the construction and analysis of large time step operator splitting algorithms for the numerical simulation of multiphase flow in heterogeneous porous media. The purpose of these lecture notes is to review some of this activity. We illustrate the main ideas behind these novel operator splitting algorithms for a basic twophase flow model. Special focus is posed on the numerical solution algorithms for the saturation equation, which is a convection dominated, degenerate convectiondiffusion equation. Both theory and applications are discussed. The general background for the reservoir flow model is reviewed, and the main features of the numerical algorithms are presented. The basic mathematical results supporting the numerical algorithms are also given. In addition, we present some results from the BV solution theory for quasilinear degenerate parabolic equations, which provides the correct ...
Dimensional Splitting With Front Tracking And Adaptive Grid Refinement
 Numer. Methods Partial Differential Equations
, 1996
"... . Front tracking in combination with dimensional splitting is analyzed as a numerical method for scalar conservation laws in two space dimensions. An analytic error bound is derived, and convergence rates based on numerical experiments are presented. Numerical experiments indicate that large CFL num ..."
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Cited by 14 (12 self)
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. Front tracking in combination with dimensional splitting is analyzed as a numerical method for scalar conservation laws in two space dimensions. An analytic error bound is derived, and convergence rates based on numerical experiments are presented. Numerical experiments indicate that large CFL numbers can be used without loss of accuracy for a wide range of problems. A new method for grid refinement is introduced. The method easily allows for dynamical changes in the grid, using, for instance, the total variation in each grid cell as a criterion for introducing new or removing existing refinements. Several numerical examples are included, highlighting the features of the numerical method. A comparison with a highresolution method confirms that dimensional splitting with front tracking is a highly viable numerical method for practical computations. 1. Introduction Dimensional splitting, or fractional steps, has been widely used to extend onedimensional numerical methods to multidim...
Unconditionally Stable Methods For HamiltonJacobi Equations
, 2000
"... We present new numerical methods for constructing approximate solutions to the Cauchy problem for HamiltonJacobi equations of the form u t + H(Dxu) = 0. The methods are based on dimensional splitting and front tracking for solving the associated (nonstrictly hyperbolic) system of conservation laws ..."
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Cited by 5 (5 self)
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We present new numerical methods for constructing approximate solutions to the Cauchy problem for HamiltonJacobi equations of the form u t + H(Dxu) = 0. The methods are based on dimensional splitting and front tracking for solving the associated (nonstrictly hyperbolic) system of conservation laws p t + DxH(p) = 0, where p = Dxu. In particular, our methods depends heavily on a front tracking method for onedimensional scalar conservation laws with discontinuous coecients. The proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as "large time step" Godunov type (or front tracking) methods. We present several numerical examples illustrating the main features of the proposed methods. We also compare our methods with several methods from the literature.
Numerical Solution Of The Polymer System By Front Tracking
, 1999
"... . The paper describes the application of front tracking to the polymer system, an example of a nonstrictly hyperbolic system. Front tracking computes piecewise constant approximations based on approximate Riemann solutions and exact tracking of waves. It is wellknown that the front tracking method ..."
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Cited by 5 (4 self)
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. The paper describes the application of front tracking to the polymer system, an example of a nonstrictly hyperbolic system. Front tracking computes piecewise constant approximations based on approximate Riemann solutions and exact tracking of waves. It is wellknown that the front tracking method may introduce a blowup of the initial total variation for initial data along the curve where the two eigenvalues of the hyperbolic system are identical. It is demonstrated by numerical examples that the method converges to the correct solution after a finite time that decreases with the discretization parameter. For multidimensional problems, front tracking is combined with dimensional splitting and numerical experiments indicate that large splitting steps can be used without loss of accuracy. Typical CFL numbers are in the range 10 to 20, and comparisons with Riemann free, highresolution methods confirm that the high efficiency of front tracking. The polymer system, coupled with an ellip...
Piecewise SelfSimilar Solutions And A Numerical Scheme For Scalar Conservation Laws
 Laws, (SIAM J. Numer. Anal
"... The solution of the Riemann problem was a building block for general Cauchy problems in conservation laws. A Cauchy problem is approximated by a series of Riemann problems in many numerical schemes. But, since the structure of the Riemann solution holds locally in time only and, furthermore, a Riema ..."
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Cited by 3 (2 self)
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The solution of the Riemann problem was a building block for general Cauchy problems in conservation laws. A Cauchy problem is approximated by a series of Riemann problems in many numerical schemes. But, since the structure of the Riemann solution holds locally in time only and, furthermore, a Riemann solution is not piecewise constant in general, there are several fundamental issues in the approach.
Front tracking for scalar balance equations
 J. Hyperbolic Differ. Equ
"... Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the f ..."
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Cited by 3 (2 self)
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Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL–condition associated with it, and it does not discriminate between stiff and nonstiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady–state solutions (or achieving them in the long time limit) with good accuracy. 1.
Unconditionally Stable Explicit Schemes for the Approximation of Conservation Laws
"... We consider explicit schemes for homogeneous conservation laws which satisfy the geometric CourantFriedrichsLewy condition in order to guarantee stability but allow a time step with CFLnumber larger than one. A brief overview over existing unconditionally stable schemes for hyperbolic conservatio ..."
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Cited by 1 (0 self)
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We consider explicit schemes for homogeneous conservation laws which satisfy the geometric CourantFriedrichsLewy condition in order to guarantee stability but allow a time step with CFLnumber larger than one. A brief overview over existing unconditionally stable schemes for hyperbolic conservation laws is provided, although the focus is on LeVeque's large time step Godunov scheme. For this scheme we explore the question of entropy consistency for the approximation of onedimensional scalar conservation laws with convex ux function and describe a possible way to extend the scheme to the twodimensional case. Numerical calculations and analytical results show that an increase of accuracy can be obtained because the error introduced by the modi ed evolution step of the large time step Godunov scheme may be less important than the error due to the projection step.