Coupled High-Order Finite Difference and Unstructured Finite Volume Methods for Earthquake Rupture Dynamics in Complex Geometries.

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1 UPTC F11040 xamensarbete 30 hp Juni 011 Coupled High-Order Finite Difference and Unstructured Finite Volume Methods for arthquake Rupture Dynamics in Complex Geometries Ossian OReilly

2 Abstract Coupled High-Order Finite Difference and Unstructured Finite Volume Methods for arthquake Rupture Dynamics in Complex Geometries Ossian OReilly Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box Uppsala Telefon: Telefax: Hemsida: The linear elastodynamic two-dimensional anti-plane stress problem, where deformations occur in only one direction is considered for one sided non-planar faults. Fault dynamics are modeled using purely velocity dependent friction laws, and applied on boundaries with complex geometry. Summation-by-parts operators and energy estimates are used to couple a high-order finite difference method with an unstructured finite volume method. The unstructured finite volume method is used near the fault and the high-order finite difference method further away from the fault where no complex geometry is present. Boundary conditions are imposed weakly on characteristic form using the simultaneous approximation term technique, allowing explicit time integration to be used. Numerical computations are performed to verify the accuracy and time stability, of the method. Handledare: Jan Nordström Ämnesgranskare: Per Lötstedt xaminator: Tomas Nyberg SSN: , UPTC F11040

3 1. ntroduction n earthquake rupture dynamics, an earthquake shear rupture can be modeled by the use of velocity weakening rate-and-state friction laws19 and a failure mechanism that invokes the shear rupture. The failure mechanism is typically a Gaussian perturbation that reduces the normal stress slightly on a small section along the fault. f the local perturbation in normal stress is large and wide enough along the fault, then the fault fails. When the fault fails, the slip velocity instantaneously jumps from a very low, more or less static state, to a dynamic state of several magnitudes larger. As a result, two short shear wave pulses are emitted, propagating in opposite directions along the fault. The width of the shear wave pulses, or shear rupture tips have been experimentally estimated to be within the scale of centimeters 1. Furthermore, the earthquake shear rupture can propagate along the fault for several kilometers. Using numerical models, it is of great importance that the shear rupture tip is well resolved or else the fault may rupture spontaneously. Another issue is that the propagating waves propagate into the far-field for long times. t is a well-known fact that in order to reduce the error growth in time for wave propagating problems, high-order schemes and high resolution should be used 14. Therefore, the numerical models are large scale, they require high resolution, large computational domains and explicit timestepping. Summation-by-parts (SBP) high-order finite difference methods (HOFDM) are well suited for this. n addition, in 13, M.H Carpenter et al. proposed that numerical schemes for time-dependent PDs should ideally be strictly stable since it was shown that just satisfying ordinary time stability may lead to nonphysical error growth in time. Strict stability prevents this by letting the semi-discretized (discretized in space only) problem dissipate energy at a slightly faster rate than the physical model. The use of SBP operators lead to an energy estimate for hyperbolic problems, and together with the SAT penalty method developed in 15 which imposes the boundary conditions weakly; it is possible to prove strict stability. n 3, J. Kozdon et al. showed how to apply the SBP HOFDM on planar fault geometries for the so called two-dimensional anti-plane stress problem. The aim of this paper is to extend that work by considering non-planar fault geometries using an unstructured finite volume method in the near-field region of the fault and the HOFDM in the far-field region. n, J Nordström et al. showed that an unstructured finite volume method (UFVM) was on SBP form. n later works the UFVM was coupled with the HOFDM to construct an accurate and stable hybrid method 4, 5, 6. Note, while the HOFDM can be extended to handle complex geometries by coordinate transforms and patching subdomains together it is not well suited for this. n earthquake dynamics, problems of complex geometry are e.g present in complex networks of branching faults. Seen from above, in figure 1 is a real example of complex geometry in earthquake dynamics, Landers fault. Other approaches involving hybrid methods for handling non-planar fault geometries in earthquake dynamics are presented in16, 17, 18.. The continuous problem.1. The anti-plane stress problem n the theory of linear elasticity, deformations of an elastic solid are assumed to be small and governed by a linear relationship between load and deformation. Also, if the 3

4 Figure 1: Geometry of Landers fault after the Landers earthquake rupture, CA 199. Data obtained by Yann Klinger (PGP). loads on the elastic solid are removed, the elastic solid is assumed to return to its original non-deformed state. These assumptions leads to an ideal material, the linear elastic solid governed by Hooke s law. Hooke s law is a constitutive equation that relates the stresses to deformations. For more detailed explanations the reader is referred to literature on continuum mechanics, e.g 1. We consider a linear elastic solid with a frictional interface. n addition, we consider anti-plane deformation. n this case Hooke s constitutive law and the principle of conservation of linear momentum yield the governing equations σ xz t = G v z x, σ yz = G v z t y, ρ v z t = σ yz x + σ xz y, (1) where σ xz and σ yz are the shear stresses in the material acting in the ẑ-direction on planes with normals ˆx and ŷ, respectively. Furthermore, v z is the particle velocity in the ẑ-direction, ρ is the density and G is the shear modulus. For our purposes we limit ourselves to study one sided friction laws. n this case, the shear traction exerted on the solid is modeled as a velocity dependent friction law given by the shear strength τ = F (v z ) which is the solid s resistance to slip, n i σ iz = F (v z ), () where n i is the outward unit normal. n this work, an analytical function is used to model the non-linear friction law F (v z ), namely F (v z ) = βsinh 1 (γv z ), β > 0, γ > 0. (3) 4

5 This friction law is derived theoretically by considering microscopic contact between materials and thermodynamic properties. More realistic friction laws are so called rateand-state friction laws, which take time dependent effects in to account, see e.g. 19. The complete rate-and-state friction laws does not add information in this paper where we aim for stable interface coupling procedures... Symmetrization To prove stability later the governing equations in (1) need to be transformed to a symmetric form. After symmetrization the governing equations can be written as q t = à q x x + à q y y, (4) ρ v z à x = 0 c s 0 c s 0 0, à y = 0 0 c s 0 0 0, q = c s G σ xz 1 G σ yz, (5) where c s = G/ρ is the shear wave speed. For detailed calculations on how to get from (1) to (4), see the Appendix..3. Non-dimensionalization The velocity (v z ) and the stresses (σ xz, σ yz ) are non-dimensionalized by the shear impedance Z = ρc s, and the parameter β in the non-linear friction law (3), which has units of stress. The non-dimensional variables are: v z = Zv z β, σ xz = σ xz β, σ yz = σ yz β. (6) The non-linear friction law in (3) is non-dimensionalized by dividing with β and writing v z = Zv z/β, F (v z) = sinh 1 ( γβ Z v z), (7) the parameter γβ is also non-dimensional. n addition, the coordinates x, y are nondimensionalized by the wave number k = π/λ, where λ is the fundamental wave length Z of the problem. The time t is non-dimensionalized by using the shear wave speed c s, x = kx, y = ky, t = kc s t. (8) The non-dimensional variables in (6) and the non-dimensional space and time coordinates in (8) are inserted into the governing symmetric system (4), which yields the non-dimensional system, q t = A q x x + A q y y, (9) v z A x = 1 0 0, A y = 0 0 0, q = σ xz, where we have dropped the superscripts. 5 σ yz

6 .4. Well-posedness Roughly speaking, a problem is well well-posed if (i) A solution exist. (ii) That solution is unique and can be bounded by the data of the problem. n this paper we assume that the solution exist and focus on (ii). For a detailed reference on well-posedness and the energy method, see 10. The energy method is obtained by introducing the L space with the inner product (u, v) = Ω ut vdxdy for real vector valued functions. The energy of q(x, y, t) in (9) is defined by q = vz + σxz + σyzdxdy, (10) We have where (9) has been used. Green s theorem, which yields Ω d q = (q, q dt t ) = q A x Ω x + A q y dxdy, (11) x The spatial derivatives, d q = dt Ω q x and q y are transformed using q T A i n i qds, (1) where A i n i = A x n x + A x n y. The unit vector, n = (n x, n y ) is the outward unit normal with respect to the boundary Ω. ds is the infinitesimal arc length of an element of Ω in the counter-clockwise direction. The characteristic variables, which are associated with the eigenvectors of A i n i are used to impose boundary conditions so that (1) becomes dissipative. To determine the characteristics we diagonalize A i n i with the eigenvector basis X and denote the diagonalized matrix Λ. This transforms (1) into d q = dt Ω w T Λwds, (13) w = X T q = w +, w, w 0 T, (14) w ± = σ iz n i v z, (15) w 0 = σ iz m i. (16) n (16) m = ( n y, n x ), the unit tangent vector. Both w + and w propagate with the shear wave speed c s, but in opposite directions. We use the convention that w + is the outgoing characteristic variable and therefore always propagates out of the domain, or in the direction of n i at any point on the exterior boundary. Similarly, the w is the ingoing characteristic variable, always propagating into the domain or in the opposite direction of n i at any point on the exterior boundary. w 0 is the zero speed characteristic variable; No special treatment of w 0 is required according to 11. 6

7 Boundary conditions can be formulated in terms of the ingoing and outgoing characteristics by considering the following linear relationship w = R(x, y)w +. (17) The coefficient R(x, y), is the reflection coefficient which can vary along the boundary. To obtain the traction free surface boundary condition (σ iz n i = 0) we use R = 1, and for the rigid wall boundary condition (v z = 0) we have R = 1. Furthermore, we can also obtain the non-reflecting boundary condition (w = 0) by R = 0. Remark The boundary condition in (17) can be modified to include boundary data g(t). The nonlinear boundary condition in () ( σ iz n i = F (v z )) can be written on characteristic form as well. t can be considered as a non-linear relationship between characteristic variables propagating into the frictional boundary w + and the characteristic variables propagating out of the frictional boundary w, w = W (w + ). (18) Proposition.1. The characteristic relation (18) which relates the ingoing characteristic variables in terms of the outgoing characteristic variables at the frictional boundary exists if the friction law F (v z ) in () is continuously differentiable and F (v z ) 1. The proof is given in 3. Proposition.. The governing system of equations (1) with the non-linear boundary condition () are well-posed if the solution exist, v z F (v z ) 0 and F (v z ) 0. The proof is given in The discrete problem We consider the computational domain Ω shown in figure consisting of the two subdomains Ω (1) and Ω (). The lower subdomain features complex geometry and a nonlinear friction law at the fault boundary. n the lower subdomain we use the unstructured finite volume method (UFVM) and discretize using an unstructured grid consisting of M nodes. n the upper subdomain we use the high order finite difference method (HOFDM) and discretize using a structured grid consisting of N x N y grid points. The superscripts (1) and () are used to distinguish between the two subdomains, where (1) refers to the lower (UFVM) subdomain, and () refers to the upper (HOFDM) subdomain. We denote the unknown solution in the subdomains by q (1) and q (). The two methods will be coupled together at the interface. For the hybrid method (HOFDM+UFVM) to be efficient, the HOFDM subdomain should be much larger than the UFVM subdomain 4. n figure this is shown by the lengths in the y-direction, L (1) y and L () y, where L () y >> L (1) y. The boundaries are defined in an overlapping manner, i.e. overlapping at the corner nodes (e.g. the exterior west boundary overlap with the interior boundary at the northwest corner) The boundary conditions are then imposed on each boundary. 7

8 North boundary, NORTH () Ly West boundary, WST () Structured grid Ω () ast boundary, AST () nterface, West boundary, WST (1) (1) Ly Unstructured grid Ω (1) ast boundary, AST (1) y () x (1) Ly >>Ly Frictional fault, FAULT Figure : The two subdomains Ω (1) and Ω () will be coupled together at the interface. The unstructured finite volume method will be used in the subdomain Ω (1) and the high order finite difference method will be used in the subdomain Ω (). 8

9 3.1. Summation-by-parts operators The numerical schemes that will be constructed uses summation-by-parts (SBP) operators that result in a discrete analogy of the continuous energy estimate. To introduce the technique we consider the 1D scalar advection problem, u t = u, x 0, 1. (19) x By multiplying with u and integrating over the domain we get A semi-discrete analogy is d dt u = 1 0 u du dx dx = u(1, t) u(0, t). (0) P v = Qv, (1) t where v is the unknown solution discretized in space with N grid points v = v 1 (t),... v N (t). The matrix Q is nearly skew-symmetric, i.e Q + Q T = () That is Q + Q T has zero entries for nodes associated with the interior and non-zero entries for nodes associated with the boundary. P is a positive semi-definite matrix, and used to define the discrete norm v P = vt Pv. (3) By multiplying (1) with v T, adding the transpose, and using (), (3) we obtain d v P dt = v N v 0. (4) Clearly, by comparing (4) with (0), the SBP operators yield integration by parts in a discrete sense. To conclude, for the exact solution u and the discrete solution v the SBP operators P, Q satisfy the following properties (i) P u x Qu = PTe, where Te is the truncation error. (ii) v T Pv > 0 for all v 0, and P = P T (iii) The matrix Q is nearly skew-symmetric (cf. ()). 9

10 3.. The node-centered finite volume method To handle complex geometry a node-centered finite volume method is used. t operators on unstructured grids with polygonal elements; in fact the unstructured grid can constitute a mixture of elements (e.g in D, triangles and quadrilaterals). Stability is guaranteed through the use of SBP operators provided that the boundary conditions are implemented in a stable manner (see section 3.3 ). On an arbitrary triangular grid the method is least first order accuracte in the interior, see 8 for proof. On very regular meshes the accuracy approaches two. Here we only consider first order differential operators, but schemes for higher order differential operators are possible, see for example 9 on how to construct the Laplacian operator. The UFVM is obtained by integrating (9) over a region Ω i Ω (1) and applying Green s theorem on the spatial derivative terms which leads to Ω i q t dxdy = A x qdy A y qdx (5) Ω i Ω i The region Ω i is the control volume around N i, where N i refers to node i in the unstructured grid, see Fig. 3. One control volume is formed around each node in the unstructured grid, and together the control volumes constitute the dual grid. ach control volume is constructed in a non-overlapping manner by using the midpoints and center of gravities of the neighboring edges and elements of the unstructured grid. A semi-discrete approximation of (5) using that technique can be written as ( 3 P) q t = (Q x A x )q + (Q y A y )q, (6) where 3 is the identity matrix of dimension 3 x 3. P, Q x and Q x are SBP operators of dimension M M satisfying the SBP properties (i)-(iii). The binary operator is the Kronecker-product defined as A B = a 00 B a 0N B..... a M0 B a MN B, (7) where the matrices A, B may be of any dimension. n (6), q is of dimension 3M 1 and is organized with the three non-dimensional variables in (9) at the first node followed by the non-dimensional variables at the next node and so on, q = q 1, q,... q M T, q i = (v z ) i, (σ xz ) i, (σ yz ) i, i = 1,,..., M. Note that no boundary conditions are imposed in (6), they will be considered in subsection 3.3. The positive semi-definite SBP operator P is obtained by approximating the first term in (5) using q t dxdy V q i i t, (8) Ω i 10

11 N i N j y j Ω i Ωi y i N i y x x j y x Ω ( a) ( b ) x i Figure 3: (a) n the interior (Ω (1) ), (b) On the boundary ( Ω (1) ). N j is a neighboring node to N i. x j and y j follow the counter-clockwise convention The solid lines comprise the unstructured grid, and the dashed lines comprise the dual grid. 11

12 where V i is the area of the control volume. ach V i is stored on the diagonal of the matrix P, off-diagonal entries are zero. The operators Q x and Q y are derived in. They are approximations of the line integral terms in (5), and are defined in the following way if i / Ω (1), (Q x ) ij = y j (Q y ) ij = x j if i Ω (1), (Q x ) ij = y j (Q y ) ij = x j = (Q x ) ji, (Q x ) ii = 0, = (Q y ) ji, (Q y ) ii = 0, = (Q x ) ji, (Q x ) ii = y i, = (Q y ) ji, (Q y ) ii = x i. x j, y j are defined in figure 3. The relations (9) yield the SBP matrices (9) X = Q x + Q T x, Y = Q y + Q T y. (30) n (30), X, Y, are diagonal and the only entries are y i, - x i ; where i is a boundary node index, i.e q T Xq = qi y i, q T Yq = qi x i. (31) i Ω (1) i Ω (1) 3.3. Characteristic boundary treatment of the UFVM We will use the simultaneous approximation term (SAT) penalty method to enforce characteristic boundary conditions weakly. n the SAT penalty method, the numerical solution is penalized for not satisfying the boundary conditions, see 15. SAT penalty terms are added to the right-hand side of the semi-discrete UFVM scheme in (6). We have dq (1) dt = D (1) x q (1) + D (1) y q (1) + FT (1) + BT (1) W + BT(1) + T(1), (3) D (1) x = (P (1) ) 1 Q x (P (1) ) 1 Q y D (1) y = FT (1) = (P (1) ) 1 (1) F BT (1) W = (P (1) ) 1 (1) W BT (1) = T (1) = A x A y Σ(1) F Σ(1) W w (1) W (w +(1) ) e 3 (w (1) R (1) W w+(1) ) e 3 (w (1) R (1) w+(1) ) e 3 (P (1) ) 1 (1) Σ(1) (P (1) ) 1 (1) (1) Y Σ (q (1) (1) m q () ), The SAT penalty terms are FT (1), BT (1) W, BT(1) and T(1), each one of dimension 3M M. The term FT (1) will enforce a non-linear friction law on the fault boundary. The terms BT (1) W and BT(1) will enforce the linear boundary conditions (17) on the ast 1

13 and West boundaries of Ω (1) ; R is the reflection coefficient matrix which is diagonal and who s elements r ij have a magnitude less than or equal to one. The coupling at the interface of the UFVM and the HOFDM is achieved through the term T (1) which we will consider in subsection 3.5. The SAT penalty terms FT (1), BT (1) W and BT(1) are all determined in a similar manner, therefore we choose to only show the procedure for FT (1). The term Σ (1) is an unknown penalty matrix of dimension 3 3 that we will determine such that the semi-discrete system is dissipating energy slightly faster than the continuous system. The vector w (1) is of dimension M 1 and corresponds to the ingoing characteristic variables. n order to project the penalty term FT (1) onto the fault boundary we use the projection matrix F. The projection matrix is of dimension M M with zeros everywhere except on the diagonal which are associated with the nodes on the fault boundary (for those elements the values are one). We also have e 3 = T. An energy estimate is obtained by multiplying (3) with (q (1) ) T P (1) 3 from the left, and then adding the transpose of that. We get d q (1) = (q (1) ) T (X A x )q (1) + (q (1) ) T (Y A y )q (1) + (33) dt P (1) 3 (q (1) ) T ( (1) F Σ(1) F w ) (1) W (1) (w +(1) ) e 3 + SAT, where the following Kronecker product rules has been used (A B)(C D) = (AC BD) and (A B) T = (A T B T ) provided that the matrix products AC and BD are defined. Note that we have ignored all other penalty terms except FT (1). The nearly skewsymmetric properties (31) of the matrices X and Y allow us to write (33) on summation form over the boundary values only. n addition, we assume that the ignored penalty terms cancel out all the unknown boundary solutions, except the solutions at the fault. The remaining terms are Observe that, d q (1) = dt P (1) 3 (q (1) j ) T A x q (1) j = (q (1) j ) T A y q (1) j = j F ault (q (1) j ) T Σ (1) F (v z (1) ) j (σ xz (1) ) j (σ (1) yz ) j (v z (1) ) j (σ (1) xz ) j (σ (1) yz ) j y j (q (1) j ) T A x q (1) j x j (q (1) j ) T A y q (1) j + (34) T w (1) j W (1) j (w +(1) j ) e (v z (1) ) j (σ (1) xz ) j = (v z (1) ) j (σ xz (1) ) j, (35) (σ yz (1) ) j (v z (1) ) j (σ (1) xz ) j = (v z (1) ) j (σ yz (1) ) j, (σ yz (1) ) j T where A x, A y are defined in (9). With the introduction of the discrete outward unit normal n j = ( y j, x j )/ s j, where s j = x j + y j together with (35), allows us 13 F

14 to write (34) as d q (1) = dt P (1) 3 j F ault (q (1) j ) T Σ (1) F s j (v (1) z ) j (n i ) j (σ (1) iz ) j+ (36) w (1) j W (1) j (w +(1) j ) e 3. We use the non-dimensionalized characteristic variable relation (15) to express the variables (v z (1) ) j and (σ (1) iz ) j in terms of characteristic variables (v z (1) ) j = w (1) j w +(1) j (37) (n i ) j (σ (1) iz ) j = y j (σ xz (1) ) j x j (σ yz (1) ) j = w (1) j + w +(1) j, s j s j (38) and inserting it into (36) yields d q (1) = dt P (1) 3 j F ault s j (q (1) j ) T Σ (1) F (w (1) j ) (w +(1) j ) + (39) w (1) j W (1) j (w +(1) j ) e 3. Similarly, the term (q (1) j ) T Σ (1) F w (1) j W (1) j (w +(1) j ) is also expressed in terms of only characteristic variables by choosing Σ (1) F in the following way: Σ (1) F = α s j 0 y j 0. (40) 0 0 x j We have (q (1) j ) T Σ (1) F e 3 = α s j (w (1) j w +(1) j ) + (w (1) j + w +(1) j ), (41) where we have used (37) and (38); α is an unknown parameter that needs to be determined. nserting the result (41) into (39) leads to d q (1) = dt P (1) 3 j F ault s j α s j w (1) j (w (1) j ) (w +(1) j ) + (4) w (1) j W (1) j (w +(1) j ). Making the choice α = 1/ and adding and subtracting (W (1) j ) gives us d q (1) = 1 dt P (1) 3 j F ault w (1) j s j (w (1) j ) (w +(1) j ) (43) w (1) j W (1) j 14 + (W (1) ) (W (1) j ) = RHS,

15 which after some algebra the right-hand side simplifies into RHS = 1 s j (w +(1) j ) (W (1) j ) + s j (w (1) j W (1) j ). j F ault (44) By using the characteristic relations (37) and (38) we get (ˆv z ) j = W (1) j w +(1) j, (n i ) j (ˆσ iz ) j = W (1) j + w +(1) j. (45) We also use (45) such that (n i ) j (ˆσ iz ) j = ˆτ = F ((ˆv z ) j ). Using these definitions we get RHS = 1 j F ault (ˆv z ) j F (ˆv z ) + s j (w (1) j W (1) j ). From the well-posedness of the continuous problem we have that v z F (v z ) 0, for all v z which leads to d q (1) 0. (46) dt P (1) 3 The results are summarized in the following proposition: Proposition 3.1. The semi-discrete system (3) with characteristic fault boundary conditions is stable with the penalty term Σ (1) F in (40) and α = 1/ The high order finite difference method n the D finite difference method the unknown solution q () of dimension 3N x N y 1 is organized as, q () = q () 11,..., q() 1N y,..., q () N x1,..., q() N xn y T (47) q () ij = (v z ) ij, (σ xz ) ij, (σ yz ) ij, i = 1,,... N x, j = 1,,... N y. A D semi-discrete approximation of the non-dimensional system of equations in (9) can be written as, q () t = D () x A x q () + D () y A y q () (48) + BT () N + BT() W + BT() + T(), BT () N = x (P (1) y ) 1 N Σ N (w () R N w +() ) e 3, (49) BT () W = (P (1) x ) 1 W y Σ W (w () R W w +() ) e 3, (50) BT () = T () = (P (1) x ) 1 y Σ (w () R w +() ) e 3, (51) x (P () y ) 1 () Σ () 15 (q () () m q (1) ), (5)

16 where D () x = (P (1) x ) 1 Q x y, D () y = x (P (1) y ) 1Q y. P (1) x, Q x are SBP operators of dimension N x N x, P (1) y, Q y are of SBP operators of dimension N y N y ; All SBP operators satisfy the SBP properties (i)-(iii) in subsection 3.1. n one dimension, the finite difference approximation of the first derivative q x (x, t) is P 1 x Q x q(t). n two dimensions q x (x, y, t) is discretized identically for every grid line in the x-direction. The penalty terms BT () N, BT() W and BT() will impose boundary conditions on the north, west and east boundaries on Ω (). R = diag((r ) i ), i = 1,,..., N x N y, is the reflection coefficient matrix for the east boundary (cf. (17)), where (R ) i 1, i = 1,,..., N x N y. (53) For the east boundary, the solution is q () = ( y 3 )q ()., = , (54) where is the projection matrix of dimension N x N x, y is the identity matrix of dimension N y N y and 3 is the identity matrix of dimension 3 3. An energy estimate is obtained in a similar way as the in the previous subsection or as shown in 3. The energy estimate becomes d q () dt = (w +() )T (P y )w +() (w () )T (P y )w () +(q () )T P y (w () R w +() ) Σ, (55) where we have neglected the penalty terms corresponding to all other boundaries except the east boundary. w +() and w () are the characteristics on the east boundary given by (15), i.e. w ±() = (σ xz ) (v z ). We choose the penalty parameter Σ in the following way: Σ = , (56) which leads to d q () dt = (w +() )T (P y R P y R )w +() (57) (w () R w +() )T P y (w () R w +() ). f (w +() )T (P y R P y R )w +() 0 then the approximation (47)-(5) is strictly stable (see 3). The stability follows from the fact that we use diagonal SBP norms, P y is diagonal and the reflection coefficient matrix R is bounded as shown in (53). 16

17 3.5. Non-characteristic interface treatment n the two previous subsections we derived penalty parameters for the enforcement of weak boundary conditions using a characteristic formulation. n this section we will work with a non-characteristic formulation to couple the two methods together. Using the UFVM scheme presented in (3) and the HOFDM scheme presented in (48) we have the complete scheme dq (1) dt dq () = D (1) x q (1) + D (1) y q (1) + T (1) + SAT (1), (58) = D () x q () + D () y q () + T () + SAT (), (59) dt T (1) = ( (1) (1) Y) Σ (q (1) q () ), T () = () x (P () y ) 1 () Σ () (q () q (1) ), SAT (1) = FT (1) + BT (1) W SAT () = BT () N + BT() W + BT(1), (60) + BT(). (61) The exterior boundary terms have already been covered and treated in subsection , and will not be considered here. The matrices (1) and () are projection matrices of dimension M M and N y N y respectively. q (1) and q () are the solutions at the interface of dimension M 1 and N x N y 1 respectively. We assume that q (1) and q () have the same organization at the interface. The case when the organizations are different is discussed in the appendix. To obtain an energy estimate we introduce the norms N (1) = (P (1) 3 ), and N () = P () x P () y for q (1) and q () respectively. After multiplying (58) by (q (1) ) T N (1) and, adding the transpose of the result we get d q (1) dt = (q(1) ) T (X A x )q (1) + (q (1) ) T (Y A y )q (1) (6) N (1) +(q (1) ) T ( (1) (1) Y) Σ (q (1) q () ) + SAT (1) = RHS (1). The solution q (1) is projected onto the interface using the projection matrix (1) and we get (q (1) ) T ( (1) Y) 3 q () = (q (1) (1) ) T (Y 3 )q (), (63) where the matrix Y is of dimension N x N x with the entries of Y corresponding to the interface. Under the assumption that the external boundary terms (60) cancel out precisely, we get RHS (1) = (q (1) ) T (Y A y )q (1) + (q (1) ) T (Y Σ (1) )(q (1) q () ). (64) A similar approach for (59) leads to d q () dt = (q() N () ) T (P () x A y )q () + (65) (q () ) T (P () x Σ () )(q () q (1) ) = RHS (). 17

18 We combine (64) and (65) to arrive at d q dt ( (1) q + ( () N (1) N ()) = (66) (q (1) ) T (Y A y )q (1) + (q (1) ) T (Y Σ (1) )(q (1) q () )+ (q () ) T (P () x A y )q () + (q () ) T (P () x Σ () )(q () q (1) ) t is straight forward to show that Y = diag(1/, 1, 1,..., 1, 1/) y, where y is the distance between two neighboring grid points on the interface. For second order SBP operators we have that P () x = Y = P x. (67) However, if the standard fourth- or sixth-order SBP norm is used this is not true. n 4 it is shown that it is possible to modify the control volumes at the interface of the UFVM scheme such that (67) still holds for standard fourth- and sixth-order SBP norms. Note that, using this modification the UFVM norm P (1), should remain unmodified for accuracy. Under the assumption that (67) holds the energy estimate becomes d q dt ( (1) q + ( () N (1) N ()) = vt Mv, (68) q (1) where v = A q (), M = P x y + Σ (1) Σ (1) Σ () Σ (1) Σ () A y + Σ (). (69) For stability we require that the matrix M is negative semi-definite. We achieve this by diagonalizing A y with Λ y = X T D A yx D, where X D is an eigenvector basis and Λ y is a diagonal matrix with the eigenvalues of A y on the diagonal. We choose the penalty matrices Σ (1), Σ () in the following way: Σ (1) = X D Λ (1) X T D and Σ() = X D Λ () X T D. We denote the diagonal components of Λ y, Λ (1) and Λ () as λ i, λ (1) i and λ () i. For the matrix M to be negative semi-definite, must hold, and that leads to We have the following proposition: λ (1) i = λ () i λ i, λ () i λ i, (70) d q dt ( (1) q + ( () N (1) N ()) = vt Mv 0. (71) Proposition 3.. The semi-discrete systems in (58) - (59) have a stable interface treatment if the conditions (67) and (70) hold. 4. Computational results Validation of the accuracy and the stability of the numerical schemes will be shown next using convergence studies and long time integrations. n the convergence studies the unstructured triangular meshes are refined using regular refinement, i.e each triangle 18

19 is split into four new triangles by inserting vertices at the midpoints of the edges. The semi-discrete systems in (58)-(59) are integrated explicitly in time using the classical fourth-order Runge-Kutta method. The time step t is chosen sufficiently small such that the error in time can be neglected in comparison to the spatial discretization errors. n the convergence studies the error is measured at a fixed time T end whereas in the long time integrations the error is measured for all discrete times. The error is monitored in the L norm in the following way: e L = q e q L, (7) where q e is the exact solution and q is the computed solution. The rate of convergence is defined as µ = Log 10( e j L / e j+1 L ) Log 10 ( N j /, (73) N j+1 ) where j refers to the grid with N j nodes (including boundary nodes). q e is constructed using the method of manufactured solutions, see e.g. 7. n the method of manufactured solutions an exact solution is chosen and then the original PD is modified by introducing source terms and choosing initial and boundary conditions based on the manufactured solution. We consider the following exact solution to (9): u z (r, t) = J 0 (r)sin(t) + v 0 t + σ 0 y, r = x + y, (74) v z = u z t, σ xz = u z x, σ yz = u z y, where u z (r, t) is the non-dimensional displacement field in the z-direction and J 0 is the Bessel function of the first kind, see figure 4. n order to satisfy the non-linear fault boundary condition in () given by the non-linear (and non-dimensional) friction law in (7) the parameter γβ/z is modified in the following way: γβ/z = sinh(n i (σ iz ) e )/(v z ) e where n i is the outward unit normal, (σ iz ) e and (v z ) e are the variables of q e. For wellposedness we require that γβ/z > 0 (cf. (3)) which we achieve by choosing the constants v 0 and σ 0 in (74) accordingly. To confirm that our computational implementation is reliable we begin by showing that the highest theoretical rate of convergence is achieved. We consider rectangular elements on a rectangular domain. The highest theoretical rate of convergence for the UFVM is O(h ) 8. n figure 5 we see that second order rate of convergence is achieved. n our second experiment we consider the non-planar fault problem, with complex geometry in the lower subdomain and a non-planar fault as shown in figure 6. The domain spans from 0, 1 0, 4.4 where the boundary of the non-planar fault is described by f(x) = 0.sin (πx) 0.05sin (3πx)sin(x 0.8). The interface is located at y = 0.4. On the interface, the spacing between the nodes is equidistant, which is easy to accomplish for any geometry and required for the stability and the accuracy of the hybrid method. We define the far-field region as 0, 1 4, 4.4. For this experiment we use v 0 = 1, σ 0 = 40, the time step t j = 0.015(0.5) j 1, j = 1,,..., 5 and the final time T end = π. We compare the hybrid method (UFVM+HODFDM) to the UFVM in the whole domain and the far-field region. Both cases (Hybrid and UFVM) use the same meshing in the lower subdomain, c.f. figure 6a, figure 6b. For the HOFDM the standard second-, fourth- 19

20 Figure 4: Bessel function of the first kind. rror Nodes rror µ Nodes Figure 5: Rate of convergence for the problem using a rectangular domain and rectangular elements. 0

21 Mesh 1 4 Ω (1) Mesh Ω (1) Ω () Mesh 3 Ω (1) y x Ω (1) f(x) =0.sin (πx) 0.05sin (3πx)sin(x 0.8) 1

22 Figure 7: xact solution (v z) e at the time t = 0.75π for the non-planar fault problem. and sixth-order SBP norms are tested. The exact solution (v z ) e at the time t = 0.75π in the whole domain is shown in figure 7. Log 10 rror Nodes Hybrid nd Hybrid 4th Hybrid 6th UFVM Nodes µ µ µ Nodes µ Figure 8: Rate of convergence for the non-planar fault problem in the whole domain. n figure 8 we see that all hybrid schemes approach the same accuracy and rate of convergence. n the far-field region, see figure 9, the errors from the lower subdomain are almost negligible. Therefore, we can see the actual rate of convergence for the HOFDM SBP operators. However, we do notice a decline in the rate of convergence for the 6th-order HOFDM SBP operator on the last two refinements. This implies that the errors from the lower subdomain are starting to become non-negligble. With further mesh refinement we can expect the rate of convergence for all HOFDM SBP operators

23 to decline as the errors from the lower subdomain begin to influence more. Log 10 rror Hybrid nd Hybrid 4th Hybrid 6th FVM Nodes Nodes µ µ µ Nodes µ Figure 9: Rate of convergence for the non-planar fault problem in the far-field region. 3

24 The absolute error of the component v z on the finest mesh is shown for all schemes in figures 10 and 11. Note that, the entire domain is shown, but image proportions are distorted for readability. As expected, we see that most of the error is in the lower subdomain for all of the hybrid schemes. Since some of the control volumes at the interface have been modified for stability reasons we can expect a less accurate result. n fact, there is a slightly larger error present in the lower subdomain for the 4th, and 6th order hybrid schemes compared to the nd order hybrid scheme. Recall that for the nd order hybrid scheme we do not need to modify any control volumes for stability. 1 x 10 4 Hybrid, nd order, Mesh 5, Absolute error v z 1 x 10 4 Hybrid, 4th order, Mesh 5, Absolute error v z x Figure 10: Absolute error at T end = π. Both grids have (0730 (UFVM) (HOFDM)) nodes. 4

25 1 x 10 4 Hybrid, 6th order, Mesh 5, Absolute error v z 1 x 10 4 UFVM, Mesh 5, Absolute error v z x Figure 11: Absolute error at T end = π. For the hybrid method the grids have (0730 (UFVM) (HOFDM)) nodes, and for the UFVM the grid has ( ) nodes. Next, we consider time stability. We use the same computational domain, compute on the first two meshes and we use the same initial variables as in the previous experiment. n this experiment, we have T end = 100π. As shown in figure 1 the error remains bounded in time for all the hybrid schemes. Log 10 rror.5 Mesh 1 Hybrid nd Hybrid 4th Hybrid 6th 3 Mesh π t 100π Figure 1: Time stability for the non-planar fault problem. 5

26 5. Conclusions and future work We have considered one sided non-planar faults for the anti-plane stress problem in complex geometries. To gain computational efficiency, UFVM and structured HOFDM were used. n the far-field region, we showed that the hybrid method is more accurate than the UFVM. The numerical scheme was constructed using SBP operators and weak imposition of boundary, and interface conditions using the SAT method. Characteristic boundary conditions were considered which allowed us to use explicit time stepping. Our computational results confirmed that the numerical scheme is stable and at least first order accurate. Future work include considering the fully elastodynamic problem. Model two sided, embedded, and branching faults. Also, incorporate rate-and-state friction laws. Appendix A. Symmetrization The governing equations in (1) are written on matrix form, u t + A u x x + A u y y = 0, A x = ρg 0 0, A y = v z 0 0 0, u = σ xz. ρ ρ ρg 0 0 σ yz (A.1) We diagonalize using the eigenvector basis P such that Λ x = P 1 A x P, where Λ x is the diagonalized A x. We multiply (A.1) with P 1 on the right-hand side and insert PP 1. We get 1 u P t + P 1 1 u Λ x PP x + P 1 1 u A y PP y = 0. (A.) Let v and à y be defined by v = P 1 u, à y = P 1 A y P. Furthermore, we assume that the system is symmetrized by some matrix S in the following way: SΛ x S 1 = (SΛ x S 1 ) T, Sà y S 1 = (Sà y S 1 ) T. After rewriting (A.) in terms of v, à y, multiplying by S and inserting S 1 S we get S v t + SΛ xs 1 S v x + Sà y S 1 S v y = 0 (A.3) Finally, define à x = SΛ x S 1, à y = Sà y S 1 and q = Sv. The symmetrized system can be written as 0 c s 0 à x = c s 0 0, q t + à q x x + à q y y = 0, ρ 0 0 c s à y = v z 0 0 0, q = 1 G σ G xz, c s = 1 ρ. c s 0 0 G σ yz (A.4) 6

27 Appendix B. Organization at the interface One way to treat the case when the solutions q (1) and q () are differently organized at the interface is to introduce mapping matrices. We have q (1) () = () m q (1), (B.1) where q (1) has the same dimension and organization as q (), but with the values of q (1) () at the interface. For non-interface entries q (1) is zero. () () m is the mapping matrix. xample: n this example we will demonstrate how the solution q (1) is mapped to q (1). Let q (1) = T, where the first two elements are the values at the interface. () n addition, let q () be of dimension 6 1 where the last two elements are associated with the interface. Then we have q (1) = T, and () () m = A scheme which takes the organization into account (excluding exterior boundary terms) can be written as dq (1) dt dq () dt = D (1) x q (1) + D (1) y q (1) + ( (1) (1) Y) Σ (q (1) m (1) q () ), = D () x q () + D () y q () + () x (P () y ) 1 () Σ () (q () m () q (1) ). The Stability proof derived in subsection 3.5 still holds since a summation can be carried out in any order. References 1 Private conversation with J. Kozdon. J. Nordström, K. Forsberg, C. Adamsson and P. liasson, Finite volume methods, unstructured meshes and strict stability for hyperbolic problems, Applied Numerical Mathematics, vol. 45, pp , J. Kozdon,. Dunham and J. Nordström, nteraction of waves with frictional interfaces using summation-by-parts difference operators: Weak enforcement of nonlinear boundary conditions, Journal of Scientific Computing, DO /s , J. Nordström and J. Gong, A stable hybrid method for hyperbolic problems, Journal of Computational Physics, vol. 1, pp , J. Gong and J. Nordström, A stable and efficient hybrid scheme for viscous problems in complex geometries, Journal of Computational Physics, vol. 6, pp , J.Nordström, F. Ham, M. Shoeybi,. van der Weide, M. Svärd, K. Mattsson, G. accarino, J. Gong, A hybrid method for unsteady inviscid fluid flow Computers & Fluids, vol. 38, pp , J. Lindström and J. Nordström, A stable and high-order accurate conjugate heat transfer problem Journal of Computational Physics, vol. 9(14),

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