SPEC & GEN RELATIVITY
SPEC & GEN RELATIVITY PHZ 6607
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Date Created: 09/18/15
Methods of Modern Numerical Relativity Benjamin Hall 9 December 2008 General Relativity Class Presentation Outline Historical Overview Areas of Concern Formulations ADM BSSN Conformal etc Constraints Gauge Conditions Gauge Shocks Boundary ConditionsExcision Recent Results Timeline for Numerical Relativity GR introduced BSSN Modern Era St 15OO hits ADM formulation anda n 1916 rd ADM l 2008 I I 3936 3956 397 3996 3906 6 Early Years 35 hits I What makes a good I formulation Stability I Hyperbolicity and wellposedness Ease of implementation Physical Meaning I Wellposed Wellposedness of a system of differential I equations guarantees existence of local solutions uniqueness of solutions continuous dependence of solutions on initial data Illposed systems may have solutions but tend towards instability independent of numerical algorithms I Hyperbolic equations Characteristic Matrix M325 fl39 If eigenvalues of M gt O Weakly Hyperbolic Not generally wellposed Eigenvalues of M gt O and M diagonalizable Strongly Hyperbolic Can be wellposed for many types M Hermitian Symmetric Hyperbolic Almost always wellposed xj 3139 dt Foliates spacetime into a series of spacelike slices reaching to spatial infinity Dynamical variables W Kquot Used by most groups from 1977 till 1999 superseded by the BSSN formulation Weakly hyperbolic and illposed Unstable even with good gauge and boundary conditions 1 R ArnowittS D eeee and CWMisnerin Gravitation39An Introduction to Current Research ed by L Witten WileyNewYork1962 I BaumgarteShapiroShibata I Nakamura BSSN 1 Strongly hyperbolic reformulation of ADM I Dynamical Variables fiiKAii1quot Significantly more stable 112logdetyij 37re4 yij Kyinij N 4 Aij e Ferry 1 T W Baumgarte and S L ShapiroPhys Rev D 59 024007 1999 I Other Formulations Conformal I Uses slices reaching to future null infinity Allows compactification and more natural boundary conditions Characteristic Foliates using characteristic cones from a central worldtube Also allows compactification and natural boundary conditions Conformal Diagram for Formulations I Conformal Slices I Constraints Equations with no time derivatives are called I constraints Analytically if the initial data obeys the constraints all future evolution will still obey Not true when numerically evolved Constraints must be met to keep solutions physical and stable I Methods for constraints I Fully constrained evolution very expensive but stable Free evolution cheap but only works shortterm Constraintdampening evolution mixed results often more stable than free evolution Importance of Constraints aDl l standard Dehweiler lq 1 39Demeiler 09051 39 Detweilerkg u l J ermr Harm 3f Hamilta ian Q nstraintj l EDD Image from Shinkai arXiv08050068v1 grqc Gauge Conditions BonaMasso slicing 1log slicing Maximal slicing Trivial shift Corotating frames Minimal distortion Gauge Shocks Coordinate pathology Characterized by discontinuities in lapse function and shift vector Caused by finite gauge propagation speeds or bad gauge choices Lara function I 153 Ii I 4 2 U E 4 B 8 1E 12 3 Taken from M Alcubierre arXiv0503030v2 grqc Boundary Conditions 31 ADMBSSN vs ConformalCharacteristic Placed at edge of computational region Constraint preserving conditions No inflow conditions Maximal Damping conditions I Singularity Excision Removes regions I from grid Usually cut at apparent horizons Involves interior boundanes Moving singularities Recent Results he Example v NE NE 39quot 200 NE0 5i EQ quot quot 39 G QNE IL 7Kquot El 3 NE 0 0 l A l quotI 150 NE 5 39I LL l I R dl I K k J 39 ME 5 I M a Iatlon IC 8 as l f 39 39 1 gt 5 I I 39 1 39 d d b M t k 100 5quot 11 l l ErEillja y I A quot39 39 24h 95 CD I I X 39 k g I quot I I IKE ll 50 3 Jl 3 I X IL I W gt39 Q V l I I 200 300 tM J Baker Astrophys J 668 2007