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Exact finite difference scheme for a spherical wave equation
Journal of Sound and Vibration (1995) 182(2), 342–344
EXACT FINITE DIFFERENCE SCHEME FOR A SPHERICAL WAVE EQUATION R. E. M Department of Physics, Clark Atlanta University, Atlanta, Georgia 30314, U.S.A. (Received 16 August 1994)
This note is concerned with the construction of an exact finite difference scheme for the spherical wave equation 1u u 1u + + = 0, 1t r 1r
u(r, 0) = f(r),
(1)
where u is a function of the radial co-ordinate r and the time t, and the initial value function f(x) has a first derivative. Such an equation can be used to model the generation of linear acoustical waves from a spherical oscillator [1–3]. The concept of an exact finite difference scheme and its implication for the numerical integration of differential equations is presented in detail by Mickins [4, 5]. Briefly, these schemes do not have numerical instabilities provided that a certain fundamental relation exists between the space and time step sizes [4]. Note that equation (1) has the exact solution u(r, t) = [g(r − t)]/r,
(2)
where g(z) has a first derivative with respect to z, but is otherwise arbitrary. With this result, the solution to the initial value problem of equation (1) is u(r, t) = [(r − t)/r]f(r − t).
(3)
The time independent form of equation (1) is dU U + = 0, dr r
U = U(r).
(4)
Its solution is U(r) = C/r,
(5)
where C is an arbitrary integration constant. For rm = (Dr)m, equation (5) becomes Um + 1 rm + 1 = Um rm ,
(6)
where Um = U(rm ). Manipulation of equation (6) allows it to be put in the form
0 1
Um + 1 − Um 1 + Um = 0. Dr rm + 1
(7)
This is the exact finite difference scheme for the ordinary differential equation (4). Now consider equation (1) without the middle term, i.e., Wt + Wr = 0.
(8)
342 0022–460X/95/170342 + 03 $08.00/0
7 1995 Academic Press Limited
343
Its exact finite difference scheme is [4, 5] Wmn + 1 − Wmn Wmn − Wmn − 1 + = 0. Dt Dr
(9)
where tn = (Dt)n, Wmn = W(rm , tn ) and it is required that the following restriction holds: Dt = Dr.
(10)
Combining the results given by the sub-equations (7) and (9), the following expressions are the exact finite difference scheme for equation (1): umn + 1 − umn umn − 1 umn − umn − 1 + + = 0, Dt rm Dr
Dt = Dr.
(11a, b)
Observe that equation (11a) with equation (11b) is equivalent to the form umn + 1 =
0 1
m−1 n um − 1 . m
(12)
A direct calculation shows that equation (12) has a solution that is equal to the corresponding discrete solution of equation (1), as given by equation (2) [6]. To verify this, we first assume that umn =
h(m − n) , rm
(13)
where h(z) is an arbitrary function of z. Substitution of this into equation (12) shows that it is a solution. Using the restriction of equation (10) or (11b) gives umn =
0 10
1
1 r − tn h m . rm Dt
(14)
0
(15)
Defining the new function q(rm − tn ) as g(rm − tn ) = h
1
rm − tn , Dt
gives umn =
g(rm − tn ) , rm
(16)
which is the discrete version of equation (2). In summary, an exact finite difference scheme has been constructed for a linear spherical wave equation. The main value of this scheme is that it allows the evaluation of the effectiveness of other finite difference schemes that are in current use for the numerical investigation of partial differential equations [7]. This research was supported in part by grants from ARO and NASA. 1. Institute for Computer Applications in Science and Engineering/Langley Research Center Workshop on Benchmark Problems 1994; Hampton, Virginia. See test set, Category 1, Problem 2. 2. R. T. B 1974 Nonlinear Acoustics. Washington, D.C.: Naval Ship Systems Command, United States Department of the Navy.
344
3. G. B. W 1974 Linear and Nonlinear Waves. New York: Wiley Interscience. 4. R. E. M 1994 Nonstandard Finite Difference Models of Differential Equations. Singapore: World Scientific. 5. R. E. M 1985 Journal of Sound and Vibration 100, 452–455. Exact finite difference schemes for the nonlinear unidirectional wave equation. 6. R. E. M 1990 Difference Equations: Theory and Applications. New York: Van Nostrand Reinhold. 7. G. D. S 1978 Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press.
EXACT FINITE DIFFERENCE SCHEME FOR A SPHERICAL WAVE EQUATION R. E. M Department of Physics, Clark Atlanta University, Atlanta, Georgia 30314, U.S.A. (Received 16 August 1994)
This note is concerned with the construction of an exact finite difference scheme for the spherical wave equation 1u u 1u + + = 0, 1t r 1r
u(r, 0) = f(r),
(1)
where u is a function of the radial co-ordinate r and the time t, and the initial value function f(x) has a first derivative. Such an equation can be used to model the generation of linear acoustical waves from a spherical oscillator [1–3]. The concept of an exact finite difference scheme and its implication for the numerical integration of differential equations is presented in detail by Mickins [4, 5]. Briefly, these schemes do not have numerical instabilities provided that a certain fundamental relation exists between the space and time step sizes [4]. Note that equation (1) has the exact solution u(r, t) = [g(r − t)]/r,
(2)
where g(z) has a first derivative with respect to z, but is otherwise arbitrary. With this result, the solution to the initial value problem of equation (1) is u(r, t) = [(r − t)/r]f(r − t).
(3)
The time independent form of equation (1) is dU U + = 0, dr r
U = U(r).
(4)
Its solution is U(r) = C/r,
(5)
where C is an arbitrary integration constant. For rm = (Dr)m, equation (5) becomes Um + 1 rm + 1 = Um rm ,
(6)
where Um = U(rm ). Manipulation of equation (6) allows it to be put in the form
0 1
Um + 1 − Um 1 + Um = 0. Dr rm + 1
(7)
This is the exact finite difference scheme for the ordinary differential equation (4). Now consider equation (1) without the middle term, i.e., Wt + Wr = 0.
(8)
342 0022–460X/95/170342 + 03 $08.00/0
7 1995 Academic Press Limited
343
Its exact finite difference scheme is [4, 5] Wmn + 1 − Wmn Wmn − Wmn − 1 + = 0. Dt Dr
(9)
where tn = (Dt)n, Wmn = W(rm , tn ) and it is required that the following restriction holds: Dt = Dr.
(10)
Combining the results given by the sub-equations (7) and (9), the following expressions are the exact finite difference scheme for equation (1): umn + 1 − umn umn − 1 umn − umn − 1 + + = 0, Dt rm Dr
Dt = Dr.
(11a, b)
Observe that equation (11a) with equation (11b) is equivalent to the form umn + 1 =
0 1
m−1 n um − 1 . m
(12)
A direct calculation shows that equation (12) has a solution that is equal to the corresponding discrete solution of equation (1), as given by equation (2) [6]. To verify this, we first assume that umn =
h(m − n) , rm
(13)
where h(z) is an arbitrary function of z. Substitution of this into equation (12) shows that it is a solution. Using the restriction of equation (10) or (11b) gives umn =
0 10
1
1 r − tn h m . rm Dt
(14)
0
(15)
Defining the new function q(rm − tn ) as g(rm − tn ) = h
1
rm − tn , Dt
gives umn =
g(rm − tn ) , rm
(16)
which is the discrete version of equation (2). In summary, an exact finite difference scheme has been constructed for a linear spherical wave equation. The main value of this scheme is that it allows the evaluation of the effectiveness of other finite difference schemes that are in current use for the numerical investigation of partial differential equations [7]. This research was supported in part by grants from ARO and NASA. 1. Institute for Computer Applications in Science and Engineering/Langley Research Center Workshop on Benchmark Problems 1994; Hampton, Virginia. See test set, Category 1, Problem 2. 2. R. T. B 1974 Nonlinear Acoustics. Washington, D.C.: Naval Ship Systems Command, United States Department of the Navy.
344
3. G. B. W 1974 Linear and Nonlinear Waves. New York: Wiley Interscience. 4. R. E. M 1994 Nonstandard Finite Difference Models of Differential Equations. Singapore: World Scientific. 5. R. E. M 1985 Journal of Sound and Vibration 100, 452–455. Exact finite difference schemes for the nonlinear unidirectional wave equation. 6. R. E. M 1990 Difference Equations: Theory and Applications. New York: Van Nostrand Reinhold. 7. G. D. S 1978 Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press.