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Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form ICERM Workshop, Brown University October 5, 2020 Kenneth Duru Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 1 / 37

The Einsteins equation � t , x i � The harmonic coordinates x µ ≡ = ( t , x , y , z ) � − gg ab ∂ b x µ � � 1 � g x µ := √− g ∂ a − = 0 , � g ab � , g ab is the inverse metric and � g denotes the wave operator. where g = det In these coordinates Einstein’s equations reduce to 10 quasilinear wave equations for the metric tensor of the form � g g ab = S ab where S ab contains the non-linear terms that do not enter the principal part of the equation, and depends only on first derivatives of the metric. Now, the scalar wave equation given by � � � 1 − gg ab ∂ b u � g u := √− g ∂ a − = 0 , with u a scalar field, has the same principal part as (2.2) and it therefore represent a fundamental model that allows us to test the numerical algorithms for the full non-linear gravitational problem. Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 2 / 37

shifted wave equation We consider the 1D shifted wave equation � ∂ u � ∂ u � � � � ∂ ∂ t − a ∂ u − ∂ ∂ t − a ∂ u + b ∂ u a = 0 , − 1 ≤ x ≤ 1 , t ≥ 0 . ∂ t ∂ x ∂ x ∂ x ∂ x Here a , b , ∈ R , with b > 0. B. Szil´ agyi, H.-O. Kreiss, and J. Winicour Phys. Rev. D 71, 104035 K. Mattsson and F . Parisi / Commun. Comput. Phys., 7 (2010), pp. 103-137 a ≡ 0: scalar wave equation (acoustics) Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 3 / 37

shifted wave equation We consider the 1D shifted wave equation � ∂ u � ∂ u � � � � ∂ ∂ t − a ∂ u − ∂ ∂ t − a ∂ u + b ∂ u a = 0 , − 1 ≤ x ≤ 1 , t ≥ 0 . ∂ t ∂ x ∂ x ∂ x ∂ x Here a , b , ∈ R , with b > 0. B. Szil´ agyi, H.-O. Kreiss, and J. Winicour Phys. Rev. D 71, 104035 K. Mattsson and F . Parisi / Commun. Comput. Phys., 7 (2010), pp. 103-137 a ≡ 0: scalar wave equation (acoustics) c = b − a 2 Solutions exhibits different characters when c > 0 and c < 0. � ∂ u � � ∂ u � ∂ ∂ ∂ t − λ 1 ∂ t − λ 2 u = 0 . ∂ x ∂ x √ √ λ 1 = a + b , λ 1 = a − b , c > 0 : λ 1 > 0 , λ 2 < 0 , c < 0 : a > 0 : λ 1 > 0 , λ 2 > 0 , a < 0 : λ 1 < 0 , λ 2 < 0 . Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 3 / 37

Energy estimate Introduce the energy � � ∂ u �� � � 2 2 � � � � ∂ t − a ∂ u ∂ u � � � � E ( t ) = + b � � � � ∂ x ∂ x We have �� �� d | ∂ a ∂ x | + | a ∂ b ∂ x | + | 1 ∂ b dt E ( t ) ≤ α 0 E ( t ) , α 0 = max ∂ t | b b The constant coefficients Cauchy problem satisfies the energy equation d dt E ( t ) = 0 . We will consider boundary conditions such that d dt E ( t ) = BTs ≤ 0 . Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 4 / 37

Stable discrete approximation ? Given: Wave equation + Initial Conditions + Boundary Conditions. Method of lines –discrete in space and continuous in time. Summation-By-Parts spectral difference operators (in space). Weak enforcement of boundary and inter-element conditions. Stability by energy methods (stable systems of ODEs). Fully discrete: Explicit Runge-Kutta. Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 5 / 37

Summation-By-Parts (SBP) Vs Integration-By-Parts (IBP) Continuous: ∂ u /∂ x � ∂ u � � � u , ∂ v ∂ x , v = − + v ( 1 ) u ( 1 ) − v ( − 1 ) u ( − 1 ) , { IBP } . ∂ x Discrete: D x u ≈ ∂ u /∂ x � D x u , v � H = −� u , D x v � H + v N u N − v 1 u 1 , { SBP } . Q + Q T = B = diag ([ − 1 , 0 , 0 , · · · , 0 , 1 ]) , H = H T > 0 . D x = H − 1 Q , Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 6 / 37

Summation-By-Parts (SBP) Vs Integration-By-Parts (IBP) Continuous: ∂ u /∂ x � ∂ u � � � u , ∂ v ∂ x , v = − + v ( 1 ) u ( 1 ) − v ( − 1 ) u ( − 1 ) , { IBP } . ∂ x Discrete: D x u ≈ ∂ u /∂ x � D x u , v � H = −� u , D x v � H + v N u N − v 1 u 1 , { SBP } . Q + Q T = B = diag ([ − 1 , 0 , 0 , · · · , 0 , 1 ]) , H = H T > 0 . D x = H − 1 Q , � � ∂ b ( x ) ∂ u Continuous: ∂ x ∂ x � ∂ � � � � � b ( x ) ∂ u b ( x ) ∂ u ∂ x , ∂ v + v ( 1 ) b ( 1 ) ∂ u ∂ x ( 1 ) − v ( − 1 ) b ( − 1 ) ∂ u , v = − ∂ x ( − 1 ) { IBP } . ∂ x ∂ x ∂ x Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 6 / 37

Summation-By-Parts (SBP) Vs Integration-By-Parts (IBP) Continuous: ∂ u /∂ x � ∂ u � � � u , ∂ v ∂ x , v = − + v ( 1 ) u ( 1 ) − v ( − 1 ) u ( − 1 ) , { IBP } . ∂ x Discrete: D x u ≈ ∂ u /∂ x � D x u , v � H = −� u , D x v � H + v N u N − v 1 u 1 , { SBP } . Q + Q T = B = diag ([ − 1 , 0 , 0 , · · · , 0 , 1 ]) , H = H T > 0 . D x = H − 1 Q , � � ∂ b ( x ) ∂ u Continuous: ∂ x ∂ x � ∂ � � � � � b ( x ) ∂ u b ( x ) ∂ u ∂ x , ∂ v + v ( 1 ) b ( 1 ) ∂ u ∂ x ( 1 ) − v ( − 1 ) b ( − 1 ) ∂ u , v = − ∂ x ( − 1 ) { IBP } . ∂ x ∂ x ∂ x Discrete: D ( b ) xx u ≈ ∂/∂ x ( b ( x ) ∂ u /∂ x ) xx u , v � P = − v T M ( b ) u T M ( b ) � D ( b ) u + v j = N ( b S x u ) j = N − v j = 1 ( b S x u ) j = 1 , u ≥ 0 , { SBP } . x x Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 6 / 37

Summation-By-Parts (SBP) Vs Integration-By-Parts (IBP) Continuous: ∂ u /∂ x � ∂ u � � � u , ∂ v ∂ x , v = − + v ( 1 ) u ( 1 ) − v ( − 1 ) u ( − 1 ) , { IBP } . ∂ x Discrete: D x u ≈ ∂ u /∂ x � D x u , v � H = −� u , D x v � H + v N u N − v 1 u 1 , { SBP } . Q + Q T = B = diag ([ − 1 , 0 , 0 , · · · , 0 , 1 ]) , H = H T > 0 . D x = H − 1 Q , � � ∂ b ( x ) ∂ u Continuous: ∂ x ∂ x � ∂ � � � � � b ( x ) ∂ u b ( x ) ∂ u ∂ x , ∂ v + v ( 1 ) b ( 1 ) ∂ u ∂ x ( 1 ) − v ( − 1 ) b ( − 1 ) ∂ u , v = − ∂ x ( − 1 ) { IBP } . ∂ x ∂ x ∂ x Discrete: D ( b ) xx u ≈ ∂/∂ x ( b ( x ) ∂ u /∂ x ) xx u , v � P = − v T M ( b ) u T M ( b ) � D ( b ) u + v j = N ( b S x u ) j = N − v j = 1 ( b S x u ) j = 1 , u ≥ 0 , { SBP } . x x � � b ( x ) ∂ u ∂ x , ∂ v ( b S x v ) j = N ≈ b ( 1 ) ∂ u ( b S x v ) j = 1 ≈ b ( − 1 ) ∂ u v T M ( b ) u ≈ , ∂ x ( 1 ) , ∂ x ( − 1 ) . x ∂ x xx = P − 1 � � P = P T > 0 . D ( b ) − M ( b ) + BS x , B = diag ( − 1 , 0 , 0 , · · · , 0 , 1 ) , x Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 6 / 37

Fully compatible SBP operators Diagonal norms H (r = 1, 2, 3, 4, 5) Interior: 2r + boundary: r . Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 7 / 37

Fully compatible SBP operators Diagonal norms H (r = 1, 2, 3, 4, 5) Interior: 2r + boundary: r . Definition Let D x and D ( b ) xx , defined above, denote SBP operators approximating ∂/∂ x , ∂/∂ x ( b ( x ) ∂/∂ x ) , with 2 r interior order of accuracy and r accurate boundary closures. Let the corresponding diagonal norm be denoted by H , P . The operators compatible SBP operators if P = H x , M ( b ) x HbD x + R ( b ) = D T x , x � � T v T R ( b ) R ( b ) R ( b ) = , x v ≥ 0 x x Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 7 / 37

Fully compatible SBP operators Diagonal norms H (r = 1, 2, 3, 4, 5) Interior: 2r + boundary: r . Definition Let D x and D ( b ) xx , defined above, denote SBP operators approximating ∂/∂ x , ∂/∂ x ( b ( x ) ∂/∂ x ) , with 2 r interior order of accuracy and r accurate boundary closures. Let the corresponding diagonal norm be denoted by H , P . The operators compatible SBP operators if P = H x , M ( b ) x HbD x + R ( b ) = D T x , x � � T v T R ( b ) R ( b ) R ( b ) = , x v ≥ 0 x x Definition The operators fully compatible SBP operators if P = H x , M ( b ) x HbD x + R ( b ) = D T x , x � � T R ( b ) R ( b ) v T R ( b ) = , x v ≥ 0 x x S x = D x . Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 7 / 37

Geometric flexibility: Duru and Virta JCP (2014) Kenneth Duru: Towards energy-stable DGSEM for Einstein’s equations of general relativity in second order form— ICERM Workshop, Brown University 8 / 37