1

CHAPTER 11

Numerical Schemes for Hyperbolic PDEs

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0; ccuu xt

])1(2)1([2

1)( 11

iiiixix uuu

xuu

1. Model Hyperbolic Equation (Linear Case)

The space derivative can be approximated by either 1st – order one-sided

differencing, or by 2nd – order central differencing (see Ch. 9, Slide 2)

Note: β = 0 → central

β = 1 → backward; leads to inherently unstable ODEs if c < 0

β = -1 → forward; leads to inherently unstable ODEs if c > 0

Since c > 0, we consider only central and backward differencing.

(11.1)

(11.2)

2

A. Explicit Schemes

(i) Explicit Euler (for time march)

For β = 0 (see FTCS; Ch. 5, Slide 8)

• O(∆t, ∆x2)

• unconditionally unstable

)(2

)(2

11111

1 ni

ni

ni

ni

ni uu

x

tcuuu

• O(∆t, ∆x2)

• conditionally stable; Courant number =

• 1-step method

])1(2)1([2

11

1ni

ni

ni

ni

ni uuu

x

c

t

uu

Lax Method: average in FTCSniu

1

x

tcC

(11.3; HC:6-6)

3

• conditionally stable;

• 1-step method

)( 11 n

ini

ni

ni uu

x

tcuu

For β = 1 (FTBS)

• O(∆t, ∆x)

• σ - λ relationship for Explicit Euler is σ = 1 +λΔt

• from Ch. 9, Slide 4:

M

mi

M

m

x

cm

2sin

2cos1

1

x

tcC

(11.4; HC:6-4)

4

(ii) β = 0, Leapfrog (for time march)

• see Ch. 5, Slides 9,10,14-28

• O(∆t2, ∆x2)

• conditionally stable;

• 2-step method

1

x

tcC

)( 1111 n

ini

ni

ni uu

x

tcuu

(11.5; HC:6-7)

5

(iii) Lax-Wendroff Method

• see Ch. 8, Slide 5 (for 2D)

• O(∆t2, ∆x2)

• conditionally stable;

• 1-step method

1

x

tcC

)2(2

)(2

112

22

111 n

ini

ni

ni

ni

ni

ni uuu

x

tcuu

x

tcuu

(11.6; HC:6-11)

Note: Before applying central space differencing, the Lax-Wendroff

technique leads to the equation

n

i

n

i

ni

ni

x

utc

x

uc

t

uu

2

221

2

artificial dissipation ↑

6

B. Implicit Schemes

(i) Implicit Euler (for time march)

])1(2)1([2

11

111

1

ni

ni

ni

ni

ni uuu

x

c

t

uu

For β = 0 (see BTCS; Ch. 6, Slide 1)

• O(∆t, ∆x2)

• unconditionally stable

• 1-step method, using tridiagonal solver

)(2

11

11

1

n

ini

ni

ni uu

x

tcuu (11.7; HC:6-12)

7

For β = 1 (BTBS)

• O(∆t, ∆x)

• σ - λ relationship for Implicit Euler is σ = (1 – λΔt)-1

• from Ch. 9, Slide 4:

M

mi

M

m

x

cm

2sin

2cos1

• unconditionally stable

• 1-step method, using tridiagonal solver

)( 11

11

n

ini

ni

ni uu

x

tcuu (11.8; HC:6-13)

8

(ii) Crank-Nicolson, with β = 0

• see Ch. 6, Slide 2

• O(∆t2, ∆x2)

• unconditionally stable

• 1-step method, using tridiagonal solver

)(4

111

11

11 n

ini

ni

ni

ni

ni uuuu

x

tcuu

(11.9; HC:6-15)

9

C. Splitting Methods

• for multidimensional problems

• Approximate Factorization – see, eg.,

Lax-Wendroff for 2D (Ch. 8, Slides 5-8)

C-N:

Factored form (Ch. 8, Slides 15,16)

Factored delta form (Ch. 8, Slides 17,18)

ADI (Ch. 8, Slide 19)

ADI (Ch. 7, Slides 12-14,16,17)

Fractional Step (Ch. 7, Slides 15,18)

10

D. Multi-Step Methods

• better suited for nonlinear problems

(i) Richtmyer/Lax-Wendroff (2-step)

Richtmyer:

• apply equations at i

• O(∆t2, ∆x2)

• conditionally stable;

• see equations in H&C: 6-17, 6-18

Step 1: n → n + ½; use Lax method

Step 2: n + ½ → n + 1; use Leapfrog method

2

x

tcC

11

Lax-Wendroff:

• apply equations at i + ½

• O(∆t2, ∆x2)

• conditionally stable;

• see equations in H&C: 6-19, 6-20

1

x

tcC

(ii) MacCormack Method

• see Ch. 5, Slides 3,4

• O(∆t2, ∆x2)

• conditionally stable;

• see equations in H&C: 6-22, 6-23

1

x

tcC

See Hoffmann & Chiang, p. 191-206 for a computational example

and comparison of methods (linear problem).

12

2. Model Hyperbolic Equation (Nonlinear Case)

x

uu

t

u

(11.10)

x

E

t

u

where E = u2/2or (11.11)

• this are referred to as the “inviscid Burgers equation”

• wave propagates with different velocity at each point

• may eventually form a discontinuity - similar to shock wave formation

from a series of compression waves

13

A. Explicit Schemes

(i) Lax Method

Applying Explicit Euler and central space differencing to (11.11):

)(2

111 n

ini

ni

ni EE

x

tuu

Stability is improved by averaging to give niu

)(2

)(2

11111

1 ni

ni

ni

ni

ni EE

x

tuuu

or, in terms of u,

14

• explicit method

• nonlinear terms appear at level n on RHS; no linearization needed

• O(Δt, Δx2)

• conditionally stable: Courant number =

• O(Δt) implies dissipative errors may occur – initial discontinuity

smeared over several grid points

• see H&C, p. 208, Fig. 6-20

1max

x

tuC

))()((4

)(2

1 2

1

2

111

1 n

i

n

i

n

i

n

i

n

i uux

tuuu

(11.12; HC:6-34)

15

(ii) Lax-Wendroff Method (see Chapter 8)

Expand in Taylor series about , and replace time

derivatives using the PDE.

Note that E = E(u) so, for example,

niu1n

iu

x

EA

x

E

u

E

t

u

u

E

t

E

where Jacobian

u

EA

Since E = u2/2, we get A = u, so the above becomes

x

Eu

t

E

Then, one can show that

x

Eu

xx

EA

xt

u2

2

16

Substituting into the Taylor series gives

)(2

21

tOt

x

EA

xx

E

t

uu ni

ni

↑ like an error term in modified PDE

Central space differencing and some averaging leads to

)(2

111 n

ini

ni

ni EE

x

tuu

)])(())([()(4

)(11112

2ni

ni

ni

ni

ni

ni

ni

ni EEuuEEuu

x

t

(11.13; HC:6-39)

- see H&C for derivation, page 210.

17

• explicit method

• nonlinear terms appear at level n on RHS; no linearization needed

• O(Δt2, Δx2)

• conditionally stable: Courant number =

• O(Δt2) implies dispersion errors may occur – oscillations near

the discontinuity

• see H&C, p. 211-212, Figs. 6-22, 6-23

1max

x

tuC

18

(iii) MacCormack Method

• predictor-corrector or multi-level type scheme

• explicit method; no linearization needed

• O(Δt2, Δx2)

• conditionally stable: Courant number =

• well-behaved; no oscillations and very little smearing when

C = 1, due to the splitting which applies a forward, then a

backward difference for the space derivative

1max

x

tuC

)( 1* n

ini

nii EE

x

tuu

)(2

1 *1

**1iii

ni

ni EE

x

tuuu (11.14; HC:6-40,6-41)

• see H&C, p. 212-213, Figs. 6-24, 6-25

19

(iv) FTBS

Applying Explicit Euler and backward space differencing to (11.11):

)( 11 n

ini

ni

ni EE

x

tuu

(11.15; HC:6-49)

• explicit method

• nonlinear terms appear at level n on RHS; no linearization needed

• O(Δt, Δx)

• conditionally stable: Courant number =

• O(Δt) implies dissipative errors may occur – initial discontinuity

smeared over several grid points; but O(Δx) upwinding introduces

artificial viscosity which may suppress these errors

• see H&C, p. 219, Fig. 6-29

1max

x

tuC

20

(i) Beam and Warming Method

• implicit method; use tridiagonal solver

• FDE is nonlinear; linearized by lagging the Jacobian

• O(Δt2, Δx2)

• unconditionally stable

• O(Δt2) implies dispersion error (very large); add 4th order

damping term (like adding artificial viscosity). Since 4th

order, it doesn’t affect the O(Δx2) accuracy.

• advantages of implicit scheme (stable, larger Δt) is lost due

to dispersion errors (large oscillations)

B. Implicit Schemes

21

4

44

111111 )(44

)(2 x

uxuA

x

tuA

x

tEE

x

tu e

ni

ni

ni

ni

ni

ni

ni

1

1111

1144

ni

ni

ni

ni

ni uA

x

tuuA

x

t

(11.16; HC:6-47,6-48)

• for stability, we must choose 0 < εe ≤ 0.125

• the damping term can be approximated by

)464()( 21124

44 n

ini

ni

ni

niee uuuuu

x

ux

• see H&C, p. 216-218, Figs. 6-26, 6-27, 6-28

22

(ii) BTBS

Applying Implicit Euler and backward space differencing to (11.11):

)( 11

11

n

ini

ni

ni EE

x

tuu

• nonlinear terms appear at level n+1 on RHS

• linearization needed – lag the value of u, i.e., write

1211

2

1)(

2

1 nnnn uuuE

23

• implicit method

• use bidiagonal solver (lower bidiagonal matrix – forward sweep)

• O(Δt, Δx)

• unconditionally stable

• see H&C, p. 220, Fig. 6-30

(11.17; HC:6-51)

This scheme can be expressed in the bidiagonal form

ni

ni

ni

ni

ni uuu

x

tuu

x

t

1111

21

2

24

C. RK Schemes

• popular explicit methods for solving ODEs; easy to program

• uses weighted average of several solutions over time interval

Δt – improves accuracy

• RK2 or 2-stage RK method – 2nd order accurate

• RK4 or 4-stage RK method – 4th order accurate; most commonly

used RK method

• Modified RK methods – reduces storage requirements

• better stability than comparable explicit schemes

• requires significantly more computations per time step

• difficult to estimate errors

• see H&C, p. 219-227.

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D. Other Schemes

• Flux Corrected Transport

• Total Variation Diminishing (TVD) Schemes

- 1st order TVD – monotone schemes

- 2nd order TVD

- often use “flux limiters”: Harten-Yee upwind limiter,

Roe-Sweby upwind limiter, Davis-Yee symmetric

limiter

• Essentially Non-Oscillatory (ENO) Schemes

• Weighted ENO (WENO) Schemes

• Compact Schemes

• see H&C, p. 233-252.