ASPECTS OF NUMERICAL DISSIPATION AND DISPERSION

ASPECTS OF NUMERICAL DISSIPATION AND DISPERSIONMODIFIED EQUATION

MODIFIED EQUATION Let us substitute Taylor-series expansion into (1) for uit+t and ui-1t as:

Substituting Eqs (3) and (4) into (1) we have

(3)(4)

(5)MODIFIED EQUATIONRearranging (5), we obtain

Note that the left-hand-side of this equation corresponds to the wave equation and the right-hand side is the T.E, which is not zero. The significance of terms in T.E can be more easily interpreted if time-derivative terms are replaced by spatial derivatives.

(6)MODIFIED EQUATION

(7)MODIFIED EQUATION Now take the partial derivative of eq (6) w.r.t x and multiply by a:

Subtracting (8) from (7) gives

(8)

(9)Eq (9) can be written in more compact form as:

In a similar manner, we can obtain the following expressions for uttt, uttx ,and uxxt :Uttt =- a3uxxx + O[t, x], u ttx =a2uxx + O[t, x]Uxxt =-auxxx +O[t, x] (11)

(10)MODIFIED EQUATION Combining eqs (6), (9),and (11) leavesut + a ux = (ax/2) ( 1-v)uxx - [a(x)2/6](2v 23v +1)uxxx + O[(x)3, (x)2 t, x(t)2 (t)3] (12)Equation (12) is called a modified equation. It is important to emphasize that the equation obtained after substitution of Taylor-series expansions i.e. equation(6) must be used to eliminate the higher-order time derivatives rather than original PDE equation (1).

MODIFIED EQUATION

MODIFIED EQUATION Each coefficient in the modified equation is obtained by adding the coefficients in the corresponding column of the table. The right-hand side of the modified equation is T.E, since it represents the difference between the original PDE and FDE. Consequently, the lowest order term on the right-hand side of modified equation gives the order of the method. If v=1, the right-hand side of the modified becomes zero and wave equation solved exactly. In this case, the upstream differencing scheme reduces to ujt+t=uj-1t . Finite-difference that exhibit this behaviour are said to satisfy the shift condition . MODIFIED EQUATION The lowest order term of the T.E in the present case contains the partial derivative uxx , which makes this term similar to the viscous term in 1-D fluid flow equations. For example the viscous term in the 1-D NS equation may be written as/x(xx) =4/3 uxx Thus when v 1, the above scheme introduces an artificial viscosity into the solution. This is called implicit artificial viscosity. There is another artificial viscosity which is purposely added to a difference scheme. Artificial viscosity tends to reduce all gradients in the solution whether physically correct or numerically induced. This effect, which is the direct result of even derivative terms in the T.E. is called dissipation. MODIFIED EQUATION Another quasi-physical effect of numerical schemes is called dispersion. This is the direct result of the odd derivative terms that appear in the T.E. As a result of dispersion, phase relations between various waves are distorted. The combined effect of dissipation and dispersion is sometimes referred to as diffusion. MODIFIED EQUATIONIn general, if the lowest order term in the T.E. contain an even derivative, the resulting solution will predominantly exhibit dissipative (amplitude) errors. On the other hand, if the leading term is an odd derivative, the resulting solution will predominantly exhibit dispersive (phase) errors.

Initial waveDissipationDispersionERROR IN TERMS OF AMPLIFICATION FACTOR AND PHASE ANGLE In terms of amplification factor and Phase angle, the error is analyzed by considering even derivatives and odd derivatives. Even derivative dominating modified equation predicted amplification factor, while odd dominating derivative modification equation predicted phase angle change. AMPLIFICATION FACTOR AND PHASE ANGLE OF UPSTREAM SCHEMEThe amplification factor of upstream differencing scheme is:G= (1-v+vcos) i(vsin)The modulus of this amplification factor,G = [ (1-v+vcos)2 + (-vsin)2 ]1/2 can be plotted for several values of v. It is clear from this plot that v must be less than or equal to 1 if the Von Neumann stability condition G 1 is to be met.AMPLIFICATION FACTOR AND PHASE ANGLE OF UPSTREAM SCHEME (1- +cos)2 (-sin)2 G 1 0 1 0 1 1 45 0.5 0.5 1 1 90 0 1 1 1 135 0.5 0.5 1 1 180 1 0.5 1 0.5 0 1 0 10.5 45 0.728 0.125 0.9230.5 90 0.25 0.25 0.7050.5 135 0.2 0.125 0.3800.5 180 0 0 01.25 0 1 0 11.25 180 1.5 0 1.5

AMPLIFICATION FACTOR AND PHASE ANGLE OF UPSTREAM SCHEME The amplification factor can also be expressed in the exponential form for a complex number:G= G eiWhere is the phase angle given by = tan-1[Im(G)/Re(G)]=tan-1[-vsin/1-v+vcos]The phase angle for the exact solution of the wave equation (e) is determined in the similar manner once the amplification factor of the exact solution is known. AMPLIFICATION FACTOR AND PHASE ANGLE OF UPSTREAM SCHEMEIn order to find the exact amplification factor we substitute the elemental solution u=eteikmx in to wave equation and find that =-ikmc which gives u=eikm(x-ct) The exact amplification factor is then Ge = u(t+t)/u(t)= [eikm[x-c(t+t)]/eikm(x-ct)]

AMPLIFICATION FACTOR AND PHASE ANGLE OF UPSTREAM SCHEMEWhich reduces toGe= e-ikmct = eieWhere e =-kmc t = =-kmx ct/ x = -v And Ge =1Thus the total dissipation (amplitude) error that accrues from applying the upstream differencing method to the wave equation for N steps is given by (1 G N)A0 where A0 is the initial amplitude of the wave.RELATIVE PHASE ERROR OF UPSTREAMDIFFERENCING SCHEME

AMPLIFICATION FACTOR AND PHASE ANGLE OF UPSTREAM SCHEME Likewise the total dispersion (phase) error can be expressed as N(e-). The relative phase shift error after one time step is given by/e=tan-1[(-vsin/1-v+vcos)]/-vFor small wave numbers (i.e small ) the relative phase error reduces to /e= 1-1/6(2v2-3v+1)2 AMPLIFICATION FACTOR AND PHASE ANGLE OF UPSTREAM SCHEMEIf the relative phase error exceeds 1 for a given value of , the corresponding Fourier component of the numerical solution has a wave speed greater than exact solution, and this is a leading phase error . If the relative phase error is less than 1, the wave speed of the numerical solution is less than the exact solution, and this is a lagging phase error. The upstream differencing scheme has a leading phase error for 0.5