WAVES 2017, Minneapolis 1

A new discontinuous Galerkin spectral element method for elastic waves withphysically motivated numerical fluxes

Kenneth Duru1,∗, Alice-Agnes Gabriel1, Heiner Igel1

1Department of Geophysics, Ludwig-Maximillian University, Munich, Germany∗Email: kenneth.duru@geophysik.uni-muenchen.de

Suggested members of the Scientific CommitteeJan Hesthaven, Julien Diaz

Abstract

The discontinuous Galerkin spectral element method(DGSEM) [1] is now an established method forcomputing approximate solutions of partial dif-ferential equations in many applications. InDGSEM, numerical fluxes [2] are used to enforceinternal and external physical boundary condi-tions. This has been successful for many prob-lems [3]. However, for certain problems suchas elastic wave propagation in complex media,and where several wave types and wave speedsare simultaneously present, a numerical flux [2]may not be compatible with physical boundaryconditions. For example if surface or interfacewaves are present, this incompatibility may leadto numerical instabilities. We present a stableand arbitrary order accurate DGSEM for elas-tic waves with physically motivated numericalfluxes. Our numerical flux is compatible with allwell-posed physical boundary conditions, andlinear and nonlinear friction laws. By construc-tion our choice of penalty parameters yield anupwind scheme and a discrete energy estimateanalogous to the continuous energy estimate.

Keywords: elastodynamics, dicontinuous Galerkin,spectral method, boundary conditions, stability

1 Model problem

Consider the 1D elastic wave equation, in 0 ≤x <∞,

ρ(x)∂v

∂t=∂σ

∂x,

1

µ(x)

∂σ

∂t=∂v

∂x, t ≥ 0, (1)

with (v(x, 0), σ(x, 0)) = (v0(x), σ0(x)). Define theleft-going p, and the right-going q characteris-tics p(v, σ) = 1

2 (Zsv + σ), q(v, σ) = 12 (Zsv − σ),

Zs = ρcs. At the boundary, x = 0, we imposethe general linear well-posed boundary condi-tions: B(v, σ) := q − rp = 0, having

B(v, σ) :=Zs2

(1− r) v − 1 + r

2σ = 0, x = 0, (2)

where r is real |r| ≤ 1. At internal boundaries,we consider a locked interface having

force balance : σ− = σ+ = σ, no slip : [[v]] = 0, (3)

where [[v]] := v+ − v−, and the superscripts-/+ denote fields at the negative and positivesides of the interface. Introduce the mechanicalenergy E(t) = 1

2

∫Ω

(ρ|v|2 + 1

µ |σ|2)dx, we have

dE(t)

dt= σ[[v]]− v(0, t)σ(0, t).

The internal term vanishes σ[[v]] = 0 and the ex-ternal boundary terms are negative semi-definite,having E(t) ≤ E(0). This energy loss throughthe boundaries is what the numerical methodshould mimic.

Our primary objective is to construct aninter-element procedure for a DGSEM approx-imation of (1) using the physical interface con-dition (3). The procedure is designed in a uni-fied manner such that numerical flux functionsare compatible with the general linear bound-ary condition (2). We will now reformulate theboundary condition (2) and interface condition(3) by introducing transformed (hat-) variablesso that we can simultaneously construct (nu-merical) boundary/interface data for particle ve-locity and traction. The hat-variables preservethe amplitude of the outgoing characteristicsand satisfy the physical boundary/interface con-ditions (2) and (3) exactly. Example, the hat-variables, v0, σ0, for the boundary at x = 0solve the linear algebraic problem: B(v0, σ0) = 0,

p(v0− v(0, t), σ0− σ(0, t)) = 0. Next we constructinterface data v−, σ−, v+, σ+. The hat-variablessatisfy the physical interface condition (3) andpreserve the amplitude of the outgoing charac-teristics, therefore they solve the algebraic prob-lem: σ− = σ+ = σ, [[v]] = 0, q(v− − v−, σ− −σ−) = 0, p(v+ − v+, σ+ − σ+) = 0.

2 The DGSEM

We now formulate the DGSEM for the IBVP(1), (2). We discretize [0, L] with K elements. In

2 WAVES 2017, Minneapolis

a reference element x ∈ [xk, xk+1] ↔ ξ ∈ [−1, 1],with ∆xk = xk+1−xk and test functions (φv(ξ), φσ(ξ)) ∈L2(−1, 1), the elemental weak form reads

∆xk2

∫ 1

−1

ρk(ξ)φv(ξ)∂vk(ξ, t)

∂tdξ =

∫ 1

−1

φv(ξ)∂σk(ξ, t)

∂ξdξ

− φv(−1)F k(−1, t)− φv(1)Gk(1, t), (4)

∆xk2

∫ 1

−1

1

µk(ξ)φσ(ξ)

∂σk(ξ, t)

∂tdξ =

∫ 1

−1

φσ(ξ)∂k(ξ, t)

∂ξdξ

+φσ(−1)

Zks (−1)F k(−1, t)− φσ(1)

Zks (1)Gk(1, t). (5)

The superscript k denotes a polynomial ap-proximation within the element ek = [xk, xk+1],eg. vk(ξ, t) =

∑N+1j=1 vkj (t)Lj(ξ) where Lj(ξ), are

the interpolating polynomials of degree N andvkj (t) are the degrees of freedom to be evolved.To couple solutions across the element bound-aries, we penalize data (hat-variables) againstincoming characteristics at the boundary. Thatis F k(−1, t) := q(vk− vk, σk− σk)

∣∣ξ=−1

, Gk(1, t) :=

p(vk − vk, σk − σk)∣∣ξ=1

. The integrals in (4)-

(5) are evaluated using Gauss quadrature rules,∑N+1i=1 f(ξi)wi ≈

∫ 1

−1f(ξ)dξ, that are exact for all

polynomial integrand f(ξ) of degree ≤ 2N − 1.For nodal polynomial bases we denote the ele-mental semi-discrete energy:

Ek(t) =∆xk

2

N+1∑j=1

(wj2

(ρkj |vkj (t)|2 +

1

µkj|σkj (t)|2

)).

The semi-discrete approximation satisfies theenergy equation,

d

dtE(t) = −

K∑k=2

ITkF −K−1∑k=1

ITkG − BT0,

with E(t) =∑K

k=1 Ek(t), and ITkG = 1Zs(1) |G

k(1, t)|2,

ITkF = 1Zs(−1) |F

k(−1, t)|2,

BT0 = 12

(Z1s (1− r0) |v1|2 + (1+r0)

Z1s|σ1|2

) ∣∣ξ=−1

. Since

ITkF , IT

kG,BT0 ≥ 0, then E(t) ≤ E(0). Note that

ITkF , IT

kG → 0 as ∆xk → 0.

3 Numerical tests

We present numerical experiments to demon-strate stability and accuracy, and extensions tohigh space dimension. We use nodal Lagrangebasis, with Gauss-Legendre-Lobatto (GLL) andGauss-Legendre (GL) quadrature rules, sepa-rately. We have chosen initial and boundaryconditions to match the exact solution

v(x, t) =1

2(sin (2π (x+ cst)) + sin (2π (x− cst))) .

We propagate the solution for 10 s and recordthe time-history of the numerical error in figure

0 0.2 0.4 0.6 0.8 1

x[km]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

v[m

/s]

numerical

exact

Solutions at t = 10

0 2 4 6 8 10

t [s]

10 -8

10 -6

10 -4

10 -2

10 0

err

or

GLL

GL

Numerical error

Figure 1: Solutions at t = 10 s and time history of

the numerical error using N = 4 polynomial degree

and K = 10 number of elements

1. We have performed numerical experimentsfor different resolutions and N ≤ 12. The errorsconverge spectrally to zero at the rate N + 1.In figure 2 we demonstrate extensions of ourmethod to higher space dimensions and makecomparisons with the Rusanov flux.

a) Rusanov ux b) Physically motivated ux

Figure 2: A 2D example. a) The Rusanov fluxshowing numerical instabilities from boundaries. b)The physically motivated flux showing stable solu-tions. The top panel are snapshots at t = 0.2 s andlower panel are at t = 10 s.

References

[1] J. Hesthaven and T. Warburton, NodalDiscontinuous Galerkin Methods: Al-gorithms, Analysis, and Applications,Springer, New York, 2008.

[2] V. V. Rusanov, Calculation of interactionof non-stationary shock waves with obsta-cles, J. Comput. Math. Phys. USSR, 1(1961), pp. 267–279.

[3] M. Dumbser, I. Peshkov, E. Romenski,O. Zanotti, High order ADER schemesfor a unified first order hyperbolic formu-lation of continuum mechanics: Viscousheat-conducting fluids and elastic solids, J.Comput. Phys. 5 (2016), pp. 824-862.