Wave Equations

B.5.1 Linear Wave Equation in 1D

The standard, linear wave equation in 1D for a functionu.x; t /reads

@²u

@t² Dc²@²u

@x² Cf .x; t /; x2.0; L/; t 2.0; T ; (B.68)

B.5 Wave Equations 441

wherecis the constant wave velocity of the physical medium inŒ0; L. The equation can also be more compactly written as

ut t Dc²uxxCf; x2.0; L/; t2.0; T : (B.69) Centered, second-order finite differences are a natural choice for discretizing the derivatives, leading to

ŒDtDtuDc²DxDxuCf ⁿ_i : (B.70) Inserting the exact solutionu_e.x; t /in (B.70) makes this function fulfill the equa- tion if we add the termR:

ŒDtDtu_eDc²DxDxu_eCf CRⁿ_i : (B.71) Our purpose is to calculate the truncation errorR. From (B.17)–(B.18) we have that

ŒDtDtu_eⁿ_i Du_e;t t.xi; tn/C 1

12u_e;t t t t.xi; tn/t²C_O.t⁴/;

when we use a notation taking into account thatu_eis a function of two variables and that derivatives must be partial derivatives. The notationu_e;t t means@²u_e=@t².

The same formula may also be applied to thex-derivative term:

ŒD_xD_xu_eⁿ_i Du_e;xx.x_i; t_n/C 1

12u_e;xxxx.x_i; t_n/x²C_O.x⁴/ : Equation (B.71) now becomes

u_e;t t C 1

12u_e;t t t t.xi; tn/t²Dc²u_e;xxCc² 1

12u_e;xxxx.xi; tn/x²Cf .xi; tn/ C_O.t⁴; x⁴/CRⁿ_i :

Becauseu_efulfills the partial differential equation (PDE) (B.69), the first, third, and fifth term cancel out, and we are left with

R_iⁿD 1

12u_{e;t t t t}.x_i; t_n/t²c² 1

12u_e;xxxx.x_i; t_n/x²C_O.t⁴; x⁴/; (B.72) showing that the scheme (B.70) is of second order in the time and space mesh spacing.

B.5.2 Finding Correction Terms

Can we add correction terms to the PDE and increase the order ofR_iⁿin (B.72)?

The starting point is

ŒD_tD_tu_eDc²D_xD_xu_eCf CCCRⁿ_i : (B.73)

From the previous analysis we simply get (B.72) again, but now withC: Rⁿ_i CC_iⁿD 1

12u_e;t t t t.xi; tn/t²c² 1

12u_e;xxxx.xi; tn/x²C_O.t⁴; x⁴/ : (B.74) The idea is to letC_iⁿcancel thet²andx²terms to makeRⁿ_i D_O.t⁴; x⁴/:

C_iⁿD 1

12u_{e;t t t t}.x_i; t_n/t²c² 1

12u_e;xxxx.x_i; t_n/x²: Essentially, it means that we add a new term

C D 1 12

u_{t t t t}t²c²u_xxxxx²

;

to the right-hand side of the PDE. We must either discretize these 4th-order deriva- tives directly or rewrite them in terms of lower-order derivatives with the aid of the PDE. The latter approach is more feasible. From the PDE we have the operator equality

@²

@t² Dc² @²

@x²; so

u_{t t t t} Dc²u_{xxt t}; u_xxxxDc²u_{t txx}:

Assuminguis smooth enough, so thatuxxt tDut txx, these relations lead to C D 1

12..c²t²x²/u_xx/_{t t}: A natural discretization is

C_iⁿD 1

12..c²t²x²/ŒD_xD_xD_tD_tuⁿ_i : Writing outŒDxDxDtDtuⁿ_i asŒDxDx.DtDtu/ⁿ_i gives

1 t²

u^nC1_iC1 2uⁿ_iC1Cuⁿ¹_iC1 x²

2u^nC1_i 2uⁿ_i Cuⁿ¹_i

x² Cu^nC1_i1 2uⁿ_i1Cuⁿ¹_i1 x²

:

Now the unknown valuesu^nC1_iC1,u^nC1_i , andu^nC1_i1 arecoupled, and we must solve a tridiagonal system to find them. This is in principle straightforward, but it results in an implicit finite difference scheme, while we had a convenient explicit scheme without the correction terms.

B.5.3 Extension to Variable Coefficients

Now we address the variable coefficient version of the linear 1D wave equation,

@²u

@t² D @

@x

.x/@u

@x

;

B.5 Wave Equations 443

or written more compactly as

ut t D.ux/x: (B.75)

The discrete counterpart to this equation, using arithmetic mean forand centered differences, reads h

D_tD_tuDD_x^xD_xuin

i : (B.76)

The truncation error is the residualRin the equation h

DtDtu_eDDx^xDxu_eCRin

i : (B.77)

The difficulty with (B.77) is how to compute the truncation error of the term ŒD_x^xD_xu_eⁿ_i.

We start by writing out the outer operator:

h

Dx^xDxu_e in

i D 1

x h

^xDxu_e in

iC¹₂ h ^xDxu_e

in i¹₂

: (B.78)

With the aid of (B.5)–(B.6) and (B.21)–(B.22) we have ŒD_xu_eⁿ_iC1

2

Du_e;x x_iC1

2; t_n C 1

24u_e;xxx x_iC1

2; t_n

x²C_O.x⁴/;

h ^x

i

iC¹₂ D

x_iC1 2

C1 8⁰⁰

x_iC1

2

x²C_O.x⁴/;

h

^xD_xu_ein iC¹₂ D

x_iC1 2

C1 8⁰⁰

x_iC1 2

x²C_O.x⁴/

u_e;x

x_iC1

2; tn

C 1 24u_e;xxx

x_iC1

2; tn

x²C_O.x⁴/ D

x_iC1

2

u_e;x

x_iC1

2; tn

C

x_iC1 2

1 24u_e;xxx

x_iC1

2; tn

x² Cu_e;x

x_iC1 2; t_n1

8⁰⁰ x_iC1

2

x²C_O.x⁴/ DŒu_e;xⁿ_iC1

2

CG_iⁿ_C1 2

x²C_O.x⁴/;

where we have introduced the short form G_iCⁿ 1

2

D 1 24u_e;xxx

x_iC1

2; tn

x_iC1 2

Cu_e;x

x_iC1

2; tn

1 8⁰⁰

x_iC1 2

: Similarly, we find that

h

^xD_xu_ein

i¹₂ DŒu_e;xⁿ_i1 2

CG_iⁿ 1 2

x²C_O.x⁴/ : Inserting these expressions in the outer operator (B.78) results in

h

D_x^xD_xu_ein

i D 1

x h

^xD_xu_ein iC¹₂ h

^xD_xu_ein i¹₂

D 1 x

Œu_e;xⁿ_iC1 2

CG_iCⁿ 1 2

x²Œu_e;xⁿ

i¹₂ Gⁿ

i¹₂x²C_O.x⁴/ DŒDxu_e;xⁿ_i CŒDxGⁿ_ix²C_O.x⁴/ :

The reason forO.x⁴/in the remainder is that there are coefficients in front of this term, sayHx⁴, and the subtraction and division byxresults inŒD_xH ⁿ_ix⁴.

We can now use (B.5)–(B.6) to express the D_x operator in ŒD_xu_e;xⁿ_i as a derivative and a truncation error:

ŒDxu_e;xⁿ_i D @

@x.xi/u_e;x.xi; tn/C 1

24.u_e;x/xxx.xi; tn/x²C_O.x⁴/ : Expressions likeŒD_xGⁿ_ix²can be treated in an identical way,

ŒDxGⁿ_ix²DGx.xi; tn/x²C 1

24Gxxx.xi; tn/x⁴C_O.x⁴/ : There will be a number of terms with thex²factor. We lump these now into O.x²/. The result of the truncation error analysis of the spatial derivative is there- fore summarized as

h

Dx^xDxu_e in

i D @

@x.xi/u_e;x.xi; tn/C_O.x²/ : After having treated theŒD_tD_tu_eⁿ_i term as well, we achieve

R_iⁿD_O.x²/C 1

12u_{e;t t t t}.x_i; t_n/t²:

The main conclusion is that the scheme is of second-order in time and space also in this variable coefficient case. The key ingredients for second order are the centered differences and the arithmetic mean for: all those building blocks feature second- order accuracy.

B.5.4 Linear Wave Equation in 2D/3D

The two-dimensional extension of (B.68) takes the form

@²u

@t² Dc² @²u

@x² C@²u

@y²

Cf .x; y; t /; .x; y/2.0; L/.0; H /; t 2.0; T ; (B.79) where nowc.x; y/ is the constant wave velocity of the physical mediumŒ0; L Œ0; H . In compact notation, the PDE (B.79) can be written

u_{t t} Dc².u_xxCu_yy/Cf .x; y; t /; .x; y/2.0; L/.0; H /; t 2.0; T ; (B.80) in 2D, while the 3D version reads

u_{t t} Dc².u_xxCu_yyCu_zz/Cf .x; y; z; t /; (B.81) for.x; y; z/2.0; L/.0; H /.0; B/andt 2.0; T .

Approximating the second-order derivatives by the standard formulas (B.17)–

(B.18) yields the scheme

ŒD_tD_tuDc².D_xD_xuCD_yD_yuCD_zD_zu/Cf ⁿ_i;j;k: (B.82)

B.6 Diffusion Equations 445

The truncation error is found from

ŒDtDtu_eDc².DxDxu_eCDyDyu_eCDzDzu_e/Cf CRⁿ_i;j;k: (B.83) The calculations from the 1D case can be repeated with the terms in they andz directions. Collecting terms that fulfill the PDE, we end up with

R_i;j;kⁿ D 1

12u_{e;t t t t}t²c² 1 12

u_e;xxxxx²Cu_e;yyyyx²Cu_e;zzzzz²

n

i;j;k

C_O.t⁴; x⁴; y⁴; z⁴/ :

(B.84) B.6 Diffusion Equations

B.6.1 Linear Diffusion Equation in 1D

The standard, linear, 1D diffusion equation takes the form

@u

@t D˛@²u

@x² Cf .x; t /; x2.0; L/; t 2.0; T ; (B.85) where˛ > 0 is a constant diffusion coefficient. A more compact form of the diffusion equation isut D˛uxxCf.

The spatial derivative in the diffusion equation,˛uxx, is commonly discretized asŒDxDxuⁿ_i. The time-derivative, however, can be treated by a variety of methods.

The Forward Euler scheme in time Let us start with the simple Forward Euler scheme:

ŒD^C_t uD˛DxDxuCf ⁿ_i :

The truncation error arises as the residualRwhen inserting the exact solutionu_ein the discrete equations:

ŒD^C_t u_eD˛D_xD_xu_eCf CRⁿ_i :

Now, using (B.11)–(B.12) and (B.17)–(B.18), we can transform the difference op- erators to derivatives:

u_e;t.x_i; t_n/C1

2u_{e;t t}.t_n/tC_O.t²/ D˛u_e;xx.x_i; t_n/C ˛

12u_e;xxxx.x_i; t_n/x²C_O.x⁴/Cf .x_i; t_n/CR_iⁿ: The termsu_e;t.xi; tn/˛u_e;xx.xi; tn/f .x_i; tn/vanish becauseu_esolves the PDE.

The truncation error then becomes Rⁿ_i D1

2u_e;t t.tn/tC_O.t²/ ˛

12u_e;xxxx.xi; tn/x²C_O.x⁴/ :

The Crank-Nicolson scheme in time The Crank-Nicolson method consists of us- ing a centered difference forutand an arithmetic average of theuxxterm:

ŒD_tu^nC

1 2

i D˛1

2

ŒD_xD_xuⁿ_i CŒD_xD_xu^nC1_i Cf^nC

1 2

i :

The equation for the truncation error is ŒDtu_e^nC

1 2

i D˛1

2

ŒDxDxu_eⁿ_i CŒDxDxu_e^nC1_i Cf^nC

1 2

i CR^nC

1 2

i :

To find the truncation error, we start by expressing the arithmetic average in terms of values at timet_nC1

2. According to (B.21)–(B.22), 1

2

ŒDxDxu_eⁿ_i CŒDxDxu_e^nC1_i DŒDxDxu_e^nC

1 2

i C1

8ŒDxDxu_e;t t^nC

1 2

i t²

C_O.t⁴/ :

With (B.17)–(B.18) we can express the difference operatorDxDxuin terms of a derivative:

ŒD_xD_xu_e^nC

1 2

i Du_e;xx x_i; t_nC1

2

C 1

12u_e;xxxx x_i; t_nC1

2

x²C_O.x⁴/ : The error term from the arithmetic mean is similarly expanded,

1

8ŒDxDxu_e;t t^nC

1 2

i t²D 1

8u_e;t txx

xi; t_nC¹

2

t²C_O.t²x²/ : The time derivative is analyzed using (B.5)–(B.6):

ŒD_tu^nC

1 2

i Du_e;t x_i; t_nC1

2

C 1 24u_{e;t t t}

x_i; t_nC1 2

t²C_O.t⁴/ : Summing up all the contributions and notifying that

u_e;t x_i; t_nC1

2

D˛u_e;xx x_i; t_nC1

2

Cf x_i; t_nC1

2

; the truncation error is given by

R^nC

1 2

i D1

8u_e;xx

xi; t_nC1

2

t²C 1 12u_e;xxxx

xi; t_nC1

2

x² C 1

24u_{e;t t t} x_i; t_nC1

2

t²C_O.x⁴/C_O.t⁴/C_O.t²x²/ :

B.6.2 Nonlinear Diffusion Equation in 1D We address the PDE

@u

@t D @

@x

˛.u/@u

@x

Cf .u/;