Generalization: Variable Wave Velocity

Project 2.6: Calculus with 1D mesh functions

2.7 Generalization: Variable Wave Velocity

2.7.2 Discretizing the Variable Coefficient

The principal idea is to first discretize the outer derivative. Define Dq.x/@u

@x;

and use a centered derivative aroundxDx_i for the derivative of: @

@x

n i

_iC1

2 _i1

2

x DŒDxⁿ_i : Then discretize

_iC1 2 Dq_iC1

2

@u

@x

n iC¹₂

q_iC1 2

uⁿ_iC1uⁿ_i

x DŒqD_xuⁿ_iC1 2

:

Similarly, _i1

2 Dq_i1 2

@u

@x

n i¹₂

q_i1 2

uⁿ_i uⁿ_i1

x DŒqDxuⁿ_i1 2

:

These intermediate results are now combined to @

@x

q.x/@u

@x

n

i

1 x²

q_iC1

2

uⁿ_iC1uⁿ_iq_i1 2

uⁿ_i uⁿ_i1

: (2.43) With operator notation we can write the discretization as

@

@x

q.x/@u

@x

n

i

ŒDx.q^xDxu/ⁿ_i : (2.44)

Do not use the chain rule on the spatial derivative term!

Many are tempted to use the chain rule on the term _@x^@ q.x/^@u_@x

, but this is not a good idea when discretizing such a term.

The term with a variable coefficient expresses the net fluxqu_x into a small volume (i.e., interval in 1D):

@

@x

q.x/@u

@x

1

x.q.xCx/u_x.xCx/q.x/u_x.x// :

Our discretization reflects this principle directly: qu_x at the right end of the cell minusqu_xat the left end, because this follows from the formula (2.43) or ŒD_x.qD_xu/ⁿ_i.

When using the chain rule, we get two termsqu_xxCq_xu_x. The typical dis- cretization is

ŒD_xqD_xuCD_2xqD_2xuⁿ_i; (2.45) Writing this out shows that it is different fromŒD_x.qD_xu/ⁿ_i and lacks the phys- ical interpretation of net flux into a cell. With a smooth and slowly varyingq.x/

the differences between the two discretizations are not substantial. However, whenq exhibits (potentially large) jumps, ŒDx.qDxu/ⁿ_i with harmonic aver- aging ofqyields a better solution than arithmetic averaging or (2.45). In the literature, the discretizationŒDx.qDxu/ⁿ_i totally dominates and very few men- tion the alternative in (2.45).

2.7.3 Computing the Coefficient Between Mesh Points Ifqis a known function ofx, we can easily evaluateq_iC1

2 simply asq.x_iC1 2/with x_iC1

2 D x_i C ¹₂x. However, in many casesc, and henceq, is only known as a discrete function, often at the mesh pointsx_i. Evaluatingqbetween two mesh pointsx_i andxiC1 must then be done byinterpolationtechniques, of which three are of particular interest in this context:

q_iC1

2 1

2.q_iCq_iC1/DŒq^x_i (arithmetic mean) (2.46) q_iC1

2 2

1 q_i C 1

q_iC1 1

(harmonic mean) (2.47) q_iC1

2 .q_iqiC1/¹⁼² (geometric mean) (2.48)

The arithmetic mean in (2.46) is by far the most commonly used averaging tech- nique and is well suited for smoothq.x/functions. The harmonic mean is often preferred whenq.x/exhibits large jumps (which is typical for geological media).

The geometric mean is less used, but popular in discretizations to linearize quadratic nonlinearities (see Sect.1.10.2for an example).

With the operator notation from (2.46) we can specify the discretization of the complete variable-coefficient wave equation in a compact way:

ŒDtDtuDDxq^xDxuCf ⁿ_i : (2.49) Strictly speaking,ŒDxq^xDxuⁿ_i DŒDx.q^xDxu/ⁿ_i.

From the compact difference notation we immediately see what kind of differ- ences that each term is approximated with. The notationq^xalso specifies that the variable coefficient is approximated by an arithmetic mean, the definition being Œq^x_iC1

2 D.qiCqiC1/=2.

Before implementing, it remains to solve (2.49) with respect tou^nC1_i : u^nC1_i D uⁿ¹_i C2uⁿ_i

C t

x 2

1

2.q_iCqiC1/.uⁿ_iC1uⁿ_i/1

2.q_i Cqi1/.uⁿ_i uⁿ_i1/ Ct²f_iⁿ:

(2.50)

2.7.4 How a Variable Coefficient Affects the Stability

The stability criterion derived later (Sect.2.10.3) readst x=c. IfcDc.x/, the criterion will depend on the spatial location. We must therefore choose at that is small enough such that no mesh cell hast > x=c.x/. That is, we must use the largestcvalue in the criterion:

tˇ x

maxx2Œ0;Lc.x/: (2.51)

The parameterˇis included as a safety factor: in some problems with a significantly varyingcit turns out that one must chooseˇ < 1to have stable solutions (ˇD0:9 may act as an all-round value).

A different strategy to handle the stability criterion with variable wave velocity is to use a spatially varying t. While the idea is mathematically attractive at first sight, the implementation quickly becomes very complicated, so we stick to a constanttand a worst case value ofc.x/(with a safety factorˇ).

2.7.5 Neumann Condition and a Variable Coefficient

Consider a Neumann condition@u=@xD0atxDLDNxx, discretized as ŒD_2xuⁿ_i Duⁿ_iC1uⁿ_i1

2x D0 ) uⁿ_iC1Duⁿ_i1;

fori DN_x. Using the scheme (2.50) at the end pointi DN_x withuⁿ_iC1 D uⁿ_i1 results in

u^nC1_i D uⁿ¹_i C2uⁿ_i C

t x

2 q_iC1

2.uⁿ_i1uⁿ_i/q_i1

2.uⁿ_i uⁿ_i1/

Ct²f_iⁿ (2.52) D uⁿ¹_i C2uⁿ_i C

t x

2

.q_iC1 2 Cq_i1

2/.uⁿ_i1uⁿ_i/Ct²f_iⁿ (2.53) uⁿ¹_i C2uⁿ_i C

t x

2

2qi.uⁿ_i1uⁿ_i/Ct²f_iⁿ: (2.54) Here we used the approximation

q_iC1 2 Cq_i1

2 Dqi C dq

dx

i

xC

d²q dx²

i

x²C Cqi

dq dx

i

xC

d²q dx²

i

x²C D2q_i C2

d²q dx²

i

x²C_O.x⁴/

2q_i: (2.55)

An alternative derivation may apply the arithmetic mean ofq_n1

2 andq_nC1 2 in (2.53), leading to the term

qi C1

2.qiC1Cqi1/

.uⁿ_i1uⁿ_i/ :

Since¹₂.qiC1Cq_i1/DqiC_O.x²/, we can approximate with2qi.uⁿ_i1uⁿ_i/for iDNxand get the same term as we did above.

A common technique when implementing@u=@x D 0boundary conditions, is to assumedq=dx D0as well. This impliesqiC1 Dqi1andqiC1=2Dqi1=2for iDNx. The implications for the scheme are

u^nC1_i D uⁿ¹_i C2uⁿ_i C

t x

2 q_iC1

2.uⁿ_i1uⁿ_i/q_i1

2.uⁿ_i uⁿ_i1/

Ct²f_iⁿ (2.56)

D uⁿ¹_i C2uⁿ_i C t

x 2

2q_i1

2.uⁿ_i1uⁿ_i/Ct²f_iⁿ: (2.57) 2.7.6 Implementation of Variable Coefficients

The implementation of the scheme with a variable wave velocityq.x/Dc².x/may assume thatqis available as an arrayq[i]at the spatial mesh points. The following loop is a straightforward implementation of the scheme (2.50):

for i in range(1, Nx):

u[i] = - u_nm1[i] + 2*u_n[i] + \

C2*(0.5*(q[i] + q[i+1])*(u_n[i+1] - u_n[i]) - \ 0.5*(q[i] + q[i-1])*(u_n[i] - u_n[i-1])) + \ dt2*f(x[i], t[n])

The coefficientC2is now defined as(dt/dx)**2, i.e.,notas the squared Courant number, since the wave velocity is variable and appears inside the parenthesis.

With Neumann conditionsux D 0at the boundary, we need to combine this scheme with the discrete version of the boundary condition, as shown in Sect.2.7.5.

Nevertheless, it would be convenient to reuse the formula for the interior points and just modify the indicesip1=i+1andim1=i-1as we did in Sect.2.6.3. Assuming dq=dxD0at the boundaries, we can implement the scheme at the boundary with the following code.

i = 0 ip1 = i+1 im1 = ip1

u[i] = - u_nm1[i] + 2*u_n[i] + \

C2*(0.5*(q[i] + q[ip1])*(u_n[ip1] - u_n[i]) - \ 0.5*(q[i] + q[im1])*(u_n[i] - u_n[im1])) + \ dt2*f(x[i], t[n])

With ghost cells we can just reuse the formula for the interior points also at the boundary, provided that the ghost values of bothuandqare correctly updated to ensureuxD0andqxD0.

A vectorized version of the scheme with a variable coefficient at internal mesh points becomes

u[1:-1] = - u_nm1[1:-1] + 2*u_n[1:-1] + \

C2*(0.5*(q[1:-1] + q[2:])*(u_n[2:] - u_n[1:-1]) - 0.5*(q[1:-1] + q[:-2])*(u_n[1:-1] - u_n[:-2])) + \ dt2*f(x[1:-1], t[n])

2.7.7 A More General PDE Model with Variable Coefficients

Sometimes a wave PDE has a variable coefficient in front of the time-derivative term:

%.x/@²u

@t² D @

@x

q.x/@u

@x

Cf .x; t / : (2.58) One example appears when modeling elastic waves in a rod with varying density, cf. (2.14.1) with%.x/.

A natural scheme for (2.58) is

Œ%D_tD_tuDD_xq^xD_xuCf ⁿ_i : (2.59) We realize that the %coefficient poses no particular difficulty, since% enters the formula just as a simple factor in front of a derivative. There is hence no need for any averaging of%. Often,%will be moved to the right-hand side, also without any difficulty:

ŒD_tD_tuD%¹D_xq^xD_xuCf ⁿ_i : (2.60)

2.7.8 Generalization: Damping

Waves die out by two mechanisms. In 2D and 3D the energy of the wave spreads out in space, and energy conservation then requires the amplitude to decrease. This effect is not present in 1D. Damping is another cause of amplitude reduction. For example, the vibrations of a string die out because of damping due to air resistance and non-elastic effects in the string.

The simplest way of including damping is to add a first-order derivative to the equation (in the same way as friction forces enter a vibrating mechanical system):

@²u

@t² Cb@u

@t Dc²@²u

@x² Cf .x; t /; (2.61) whereb 0is a prescribed damping coefficient.

A typical discretization of (2.61) in terms of centered differences reads

ŒD_tD_tuCbD_2tuDc²D_xD_xuCf ⁿ_i : (2.62)

Writing out the equation and solving for the unknownu^nC1_i gives the scheme u^nC1_i D

1C1

2bt

1 1

2bt1

uⁿ¹_i C2uⁿ_i CC²

uⁿ_iC12uⁿ_i Cuⁿ_i1Ct²f_iⁿ

!

; (2.63)

fori 2 _Ixⁱ andn 1. New equations must be derived foru¹_i, and for boundary points in case of Neumann conditions.

The damping is very small in many wave phenomena and thus only evident for very long time simulations. This makes the standard wave equation without damp- ing relevant for a lot of applications.