Techniques for Locally Adaptive Overview ODE Methods Timestepping - PowerPoint PPT Presentation

History of Local Timestepping Martin J. Gander Techniques for Locally Adaptive Overview ODE Methods Timestepping Developped over the Last Multirate Multirate Two Decades Extrapolation PDE Methods Hyperbolic Parabolic Space-Time FEM Martin J. Gander One-Way, Two-Way Waveform Relaxation martin.gander@unige.ch Outlook University of Geneva DD20, February 2010 Joint work with Laurence Halpern

History of Local Overview Timestepping Martin J. Gander ◮ Methods from the ODE community: Overview ◮ Split Runge-Kutta methods ODE Methods Multirate ◮ Multirate Methods Multirate Extrapolation ◮ Multirate Extrapolation Methods PDE Methods Hyperbolic Parabolic ◮ Methods from the PDE community: Space-Time FEM One-Way, Two-Way ◮ Hyperbolic Problems Waveform Relaxation Outlook ◮ Interpolation based ◮ Energy conservation ◮ Symplectic methods ◮ Parabolic Problems ◮ Explicit-Implicit ◮ Fully Implicit ◮ Space-Time Finite Elements ◮ One-Way and Two-Way methods ◮ Schwarz Waveform Relaxation

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ PDE Methods = c ( y , z , t ) Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation Outlook approximately the same amount of computer time as the prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.”

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ = c ( y , z , t ) PDE Methods Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation approximately the same amount of computer time as the Outlook prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.” Fixed known stepsizes h and H : fastest-first empty y z t H 0 h

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ = c ( y , z , t ) PDE Methods Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation approximately the same amount of computer time as the Outlook prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.” Fixed known stepsizes h and H : fastest-first empty y z t H 0 h

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ = c ( y , z , t ) PDE Methods Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation approximately the same amount of computer time as the Outlook prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.” Fixed known stepsizes h and H : fastest-first predictor y z t H 0 h

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ = c ( y , z , t ) PDE Methods Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation approximately the same amount of computer time as the Outlook prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.” Fixed known stepsizes h and H : fastest-first predictor y z t H 0 h

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ = c ( y , z , t ) PDE Methods Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation approximately the same amount of computer time as the Outlook prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.” Fixed known stepsizes h and H : fastest-first coarse y z t H 0 h

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ = c ( y , z , t ) PDE Methods Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation approximately the same amount of computer time as the Outlook prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.” Automatic variable stepsizes h and H : slowest-first empty y z t H 0 h

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ = c ( y , z , t ) PDE Methods Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation approximately the same amount of computer time as the Outlook prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.” Automatic variable stepsizes h and H : slowest-first reject y z t H 0

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ = c ( y , z , t ) PDE Methods Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation approximately the same amount of computer time as the Outlook prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.” Automatic variable stepsizes h and H : slowest-first accept y z t 0 H

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ = c ( y , z , t ) PDE Methods Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation approximately the same amount of computer time as the Outlook prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.” Automatic variable stepsizes h and H : slowest-first reject y z t 0 h H

History of Local Multirate Methods Timestepping Gear, Wells (1984): Multirate linear multistep methods Martin J. Gander Partitioning of the system of ODEs into slow and fast Overview components: ODE Methods Multirate y ′ = b ( y , z , t ) Multirate Extrapolation z ′ = c ( y , z , t ) PDE Methods Hyperbolic Parabolic “Hence the values of y will have to be approximated by Space-Time FEM One-Way, Two-Way interpolation from mesh values of y . This process will cost Waveform Relaxation approximately the same amount of computer time as the Outlook prediction step in a predictor-corrector process, because the prediction process is a linear combination of past values.” Automatic variable stepsizes h and H : slowest-first accept y z t 0 h H