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Super-Time-Stepping P. D. Mullen UIUC Email: pmullen2@illinois.edu - PowerPoint PPT Presentation

Super-Time-Stepping P. D. Mullen UIUC Email: pmullen2@illinois.edu GitHub: pdmullen 1 <latexit


  1. Super-Time-Stepping P. D. Mullen UIUC Email: pmullen2@illinois.edu GitHub: pdmullen 1

  2. <latexit sha1_base64="8mZLYPYFUvxSalRrg3ytcVMFkqE=">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</latexit> <latexit sha1_base64="8mZLYPYFUvxSalRrg3ytcVMFkqE=">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</latexit> <latexit sha1_base64="8mZLYPYFUvxSalRrg3ytcVMFkqE=">ACO3icbVBNTxsxFPTyTUohtEcuFhFST9FuVAmOCHLoqYKkEjZJPI6b8HCu7bst4jI2v/VS/9Eb1x6YGq4sodJ+wBEkayNJp5T8ziZbCYhjeB0vLK6tr6xubtQ9bH7d36rufLq0qDIcOV1KZXsIsSJFDBwVK6GkDLEskdJOb06nfvQVjhcovcKJhkLGrXKSCM/TSqP4jboNERnHkYoQ7NJnTzLBEScHLksbaKI2Kxqlh3FWzd8NW6drzZlS678MWbZejeiNshjPQRJVpEqnI3qv+Ox4kUGOXLJrO1HocaBYwYFl1DW4sKCZvyGXUHf05xlYAdulr2kB14Z01QZ/3KkM/X1hmOZtZMs8ZMZw2s703F97x+genRwIlcFwg5fzmUFpL6wNMi6VgY4CgnjBuhP8r5de+Oo6+7povIZqPvEguW80obEbnXxvHJ1UdG2SP7JMvJCKH5Jh8I2ekQzj5Sf6QB/Iv+BX8Df4Hjy+jS0G185m8QfD0DPmzryo=</latexit> <latexit sha1_base64="8mZLYPYFUvxSalRrg3ytcVMFkqE=">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</latexit> Explicit Integration: ∆ t parabolic ∝ ∆ x 2 1 ∝ N 2 D D For a 3-D problem, doubling the resolution yields: - 2 3 more zones (factor of 8 increase in cost) -time-step reduced by 1/4 (factor of 4 increase in cost) Total: 32x more expensive (!!!) 2

  3. Runge-Kutta Legendre (RKL) Super-Time-Stepping: python configure.py -sts (1) The scheme is formulated using Legendre polynomials as stability polynomials, coupled with their recurrence relations. For more details, see Meyer et al. (2014). (2) Operator split diffusive physics. (3) In operator split, execute an RKL super-time-step. The super-time-step is taken in s -stages. (4) The scheme is explicit, i.e., the j- th stage only uses solution vectors from previous stages. 3 Meyer, C. D., Balsara, D. S., & Aslam, T. 2014, JCP , 257, 594

  4. <latexit sha1_base64="+v6+TIvfQZbQWNT+Eui6Wt7msuA=">ACMHicbVBNS8NAEN34WetX1KOXxSIQkmKoBehqKDHSq0KTSyb7USXbj7YnYgl5Cd58afoRUERr/4Kt7UHtT4YeLw3w8y8IJVCo+O8WBOTU9Mzs6W58vzC4tKyvbJ6rpNMcWjxRCbqMmAapIihQIlXKYKWBRIuAh6hwP/4haUFkl8hv0U/IhdxyIUnKGROvaxdwQSGcVO7iHcoYry5lmzKOg+HXdSpliQSMGN74WK8Vxf1bZ1kdeKjl1xqs4QdJy4I1IhIzQ69qPXTXgWQYxcMq3brpOinzOFgksoyl6mIW8x6hbWjMItB+Pny4oJtG6dIwUaZipEP150TOIq37UWA6I4Y3+q83EP/z2hmGe34u4jRDiPn3ojCTFBM6SI92hQKOsm8I40qYWym/MalwNBmXTQju35fHyXmt6jpV93SnUj8YxVEi62SDbBGX7JI6OSEN0iKc3JMn8krerAfr2Xq3Pr5bJ6zRzBr5BevzC/bBqsI=</latexit> <latexit sha1_base64="+v6+TIvfQZbQWNT+Eui6Wt7msuA=">ACMHicbVBNS8NAEN34WetX1KOXxSIQkmKoBehqKDHSq0KTSyb7USXbj7YnYgl5Cd58afoRUERr/4Kt7UHtT4YeLw3w8y8IJVCo+O8WBOTU9Mzs6W58vzC4tKyvbJ6rpNMcWjxRCbqMmAapIihQIlXKYKWBRIuAh6hwP/4haUFkl8hv0U/IhdxyIUnKGROvaxdwQSGcVO7iHcoYry5lmzKOg+HXdSpliQSMGN74WK8Vxf1bZ1kdeKjl1xqs4QdJy4I1IhIzQ69qPXTXgWQYxcMq3brpOinzOFgksoyl6mIW8x6hbWjMItB+Pny4oJtG6dIwUaZipEP150TOIq37UWA6I4Y3+q83EP/z2hmGe34u4jRDiPn3ojCTFBM6SI92hQKOsm8I40qYWym/MalwNBmXTQju35fHyXmt6jpV93SnUj8YxVEi62SDbBGX7JI6OSEN0iKc3JMn8krerAfr2Xq3Pr5bJ6zRzBr5BevzC/bBqsI=</latexit> <latexit sha1_base64="+v6+TIvfQZbQWNT+Eui6Wt7msuA=">ACMHicbVBNS8NAEN34WetX1KOXxSIQkmKoBehqKDHSq0KTSyb7USXbj7YnYgl5Cd58afoRUERr/4Kt7UHtT4YeLw3w8y8IJVCo+O8WBOTU9Mzs6W58vzC4tKyvbJ6rpNMcWjxRCbqMmAapIihQIlXKYKWBRIuAh6hwP/4haUFkl8hv0U/IhdxyIUnKGROvaxdwQSGcVO7iHcoYry5lmzKOg+HXdSpliQSMGN74WK8Vxf1bZ1kdeKjl1xqs4QdJy4I1IhIzQ69qPXTXgWQYxcMq3brpOinzOFgksoyl6mIW8x6hbWjMItB+Pny4oJtG6dIwUaZipEP150TOIq37UWA6I4Y3+q83EP/z2hmGe34u4jRDiPn3ojCTFBM6SI92hQKOsm8I40qYWym/MalwNBmXTQju35fHyXmt6jpV93SnUj8YxVEi62SDbBGX7JI6OSEN0iKc3JMn8krerAfr2Xq3Pr5bJ6zRzBr5BevzC/bBqsI=</latexit> <latexit sha1_base64="+v6+TIvfQZbQWNT+Eui6Wt7msuA=">ACMHicbVBNS8NAEN34WetX1KOXxSIQkmKoBehqKDHSq0KTSyb7USXbj7YnYgl5Cd58afoRUERr/4Kt7UHtT4YeLw3w8y8IJVCo+O8WBOTU9Mzs6W58vzC4tKyvbJ6rpNMcWjxRCbqMmAapIihQIlXKYKWBRIuAh6hwP/4haUFkl8hv0U/IhdxyIUnKGROvaxdwQSGcVO7iHcoYry5lmzKOg+HXdSpliQSMGN74WK8Vxf1bZ1kdeKjl1xqs4QdJy4I1IhIzQ69qPXTXgWQYxcMq3brpOinzOFgksoyl6mIW8x6hbWjMItB+Pny4oJtG6dIwUaZipEP150TOIq37UWA6I4Y3+q83EP/z2hmGe34u4jRDiPn3ojCTFBM6SI92hQKOsm8I40qYWym/MalwNBmXTQju35fHyXmt6jpV93SnUj8YxVEi62SDbBGX7JI6OSEN0iKc3JMn8krerAfr2Xq3Pr5bJ6zRzBr5BevzC/bBqsI=</latexit> Why do all of this? The RKL scheme allows us to take a time-step that is ~s 2 times larger than the parabolic time-step! s 2 + s ∆ t STS = ∆ t parabolic 2 The desired super-time-step is the hyperbolic time-step, therefore, the ratio of the hyperbolic time-step to the parabolic time-step gives the number of stages. 4

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