Multiresolution Schemes for the Reactive Euler Equations
We present multiresolution (MR) schemes for the efficient numerical solution of the one-dimensional system of the reactive Euler equations, which has possibly stiff source terms. The original version of the method was developed by A. Harten (1995, Comm. Pure Appl. Math.48(12), 1305) for homogeneous...
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Published in | Journal of computational physics Vol. 154; no. 1; pp. 197 - 230 |
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Format | Journal Article |
Language | English |
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Elsevier Inc
01.09.1999
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Abstract | We present multiresolution (MR) schemes for the efficient numerical solution of the one-dimensional system of the reactive Euler equations, which has possibly stiff source terms. The original version of the method was developed by A. Harten (1995, Comm. Pure Appl. Math.48(12), 1305) for homogeneous hyperbolic conservation laws. By computing the cell average MR-representation of the solution, we obtain much information about the solution's regularity. This description of smoothness is then used to reduce the number of direct flux computations as well as the expensive high-order ENO (essentially nonoscillatory) reconstruction both of which are now performed only near discontinuities. Thereby, the numerical solution procedure becomes considerably more efficient. In the present case of the reactive Euler equations, the average efficiency factor measured by counting the number of actual flux computations ranges from about 5 to 12. This is on the same order of, and in some cases comes reasonably close to, actual speed-up factors obtained by code timings, which were between 3 to 5. The MR overhead rate was about 10% for the ENO and 36% for TVD schemes, respectively. The quality of the solution is shown to be the same as that of the finest grid. Detailed numerical and performance results are shown for up to fourth-order accuracy, for source terms ranging from moderate to extremely stiff. |
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AbstractList | We present multiresolution (MR) schemes for the efficient numerical solution of the one-dimensional system of the reactive Euler equations, which has possibly stiff source terms. The original version of the method was developed by A. Harten (1995, Comm. Pure Appl. Math.48(12), 1305) for homogeneous hyperbolic conservation laws. By computing the cell average MR-representation of the solution, we obtain much information about the solution's regularity. This description of smoothness is then used to reduce the number of direct flux computations as well as the expensive high-order ENO (essentially nonoscillatory) reconstruction both of which are now performed only near discontinuities. Thereby, the numerical solution procedure becomes considerably more efficient. In the present case of the reactive Euler equations, the average efficiency factor measured by counting the number of actual flux computations ranges from about 5 to 12. This is on the same order of, and in some cases comes reasonably close to, actual speed-up factors obtained by code timings, which were between 3 to 5. The MR overhead rate was about 10% for the ENO and 36% for TVD schemes, respectively. The quality of the solution is shown to be the same as that of the finest grid. Detailed numerical and performance results are shown for up to fourth-order accuracy, for source terms ranging from moderate to extremely stiff. Solution adaptive techniques provide a useful tool for the reactive Euler equations and for stiff equations in general. Here, a method based on finite volume multiresolution schemes is proposed with the goal of obtaining a high-order numerical solution on a fine grid everywhere. This is achieved by employing multiresolution techniques to adaptively select regions where the full ENO fluxes are computed and interpolate from coarser grids in the rest of the domain. In this way, adequate numerical resolution is provided at a reduced computational cost. (AIAA) |
Author | Schwendeman, Donald Bihari, Barna L |
Author_xml | – sequence: 1 givenname: Barna L surname: Bihari fullname: Bihari, Barna L organization: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York, 12180-3590, f1biharb@rpi.eduf1 – sequence: 2 givenname: Donald surname: Schwendeman fullname: Schwendeman, Donald organization: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York, 12180-3590, f3schwendd@rpi.eduf3 |
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Cites_doi | 10.1006/jcph.1994.1199 10.1016/0021-9991(83)90136-5 10.1137/0907073 10.1137/S1064827594278848 10.1016/0377-0427(94)00070-1 10.1002/cpa.3160481201 10.1016/0021-9991(90)90097-K 10.1016/0021-9991(87)90031-3 10.1080/713665233 10.1016/0168-9274(93)90117-A 10.1137/0724022 10.1006/jcph.1996.0017 |
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References | Bihari, Harten (RF2) 1995; 61 Xu, Aslam, Stewart (RF14) 1997; 1 A. K. Kapila, private communications. Harten (RF9) 1983; 49 Colella, Majda, Roytburd (RF4) 1986; 7 LeVeque, Yee (RF13) 1990; 86 Harten, Osher (RF10) 1987; 24 Bihari, Harten (RF3) 1997; 18 Harten (RF6) 1994; 115 Harten (RF7) 1993; 12 Gottschlich-Müller, Müller (RF5) June 1996 Harten, Engquist, Osher, Chakravarthy (RF11) 1987; 71 Bihari (RF1) 1996; 123 Harten (RF8) 1995; 48 Harten (10.1006/jcph.1999.6312_RF7) 1993; 12 Bihari (10.1006/jcph.1999.6312_RF1) 1996; 123 Harten (10.1006/jcph.1999.6312_RF6) 1994; 115 Harten (10.1006/jcph.1999.6312_RF8) 1995; 48 Xu (10.1006/jcph.1999.6312_RF14) 1997; 1 Bihari (10.1006/jcph.1999.6312_RF3) 1997; 18 Harten (10.1006/jcph.1999.6312_RF11) 1987; 71 10.1006/jcph.1999.6312_RF12 Colella (10.1006/jcph.1999.6312_RF4) 1986; 7 Bihari (10.1006/jcph.1999.6312_RF2) 1995; 61 Gottschlich-Müller (10.1006/jcph.1999.6312_RF5) 1996 Harten (10.1006/jcph.1999.6312_RF9) 1983; 49 LeVeque (10.1006/jcph.1999.6312_RF13) 1990; 86 Harten (10.1006/jcph.1999.6312_RF10) 1987; 24 |
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Snippet | We present multiresolution (MR) schemes for the efficient numerical solution of the one-dimensional system of the reactive Euler equations, which has possibly... Solution adaptive techniques provide a useful tool for the reactive Euler equations and for stiff equations in general. Here, a method based on finite volume... |
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Title | Multiresolution Schemes for the Reactive Euler Equations |
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