Indian Institute of Space Science and Technology
Thiruvananthapuram
AE468 Computational Fluid Dynamics
CFD Assignment 2
RIYAZ AHMED [SC20B047]
Department of Aerospace Engineering
March 2023
I.
Answer 1
The linear first-order wave equation describes the propagation of a wave in one spatial dimension. In this
case, the equation represents a wave with a constant wave speed, ”a,” that travels in the positive x-direction.
The equation implies that the rate of change of the function ”u” with respect to time, du/dt, is balanced by
the rate of change of ”u” with respect to space, du/dx, multiplied by the wave speed, ”a”.
2
The initial condition, u(x, 0) = F (x) = e−4(x−5) , specifies the initial shape of the wave at time t=0. The
function ”F(x)” is a Gaussian distribution centered at x=5 with a width of 0.5.
A.
FTCS
At t=4, the graph is showing the initial Gaussian shape propagating to the right, maintaining its Gaussian
form, but with some numerical dispersion and oscillations. The numerical errors causing the wave to lose
some amplitude, and its width is slightly larger than expected due to numerical dispersion.
At t=10, the numerical errors are more significant, and the graph might show some oscillations and numerical instability. The wave’s maximum value is significantly lower than the exact solution due to numerical
dispersion, and the wave’s width is significantly larger than expected.
Overall, the graph of the linear first-order wave equation using the FTCS method is showing some
numerical instability and errors, particularly as time progresses. The resulting wave has lost amplitude and
has a larger width due to numerical dispersion, and there is instability in the numerical solution.
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Figure 1.
Figure 2.
Figure 3. FTCS Method
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B.
Lax–Friedrichs method
At t=4, the graph shows the initial Gaussian shape propagating to the right, maintaining its Gaussian form
with reduced numerical dispersion. The wave’s amplitude is slightly lower than the exact solution due to
numerical damping, but the width of the wave is close to the exact solution.
At t=10, the graph shows the wave’s propagation further to the right with reduced numerical dispersion
compared to the FTCS method. The Gaussian shape is still recognizable, but its width might be slightly
larger than the exact solution due to numerical damping. The wave’s amplitude is still lower than the exact
solution, but less significantly compared to the FTCS method.
Overall, the graph of the linear first-order wave equation using the Lax-Friedrichs method shows a stable
numerical solution with reduced numerical dispersion compared to the FTCS method. The resulting wave
might have a slightly lower amplitude and a larger width due to numerical damping, but the Gaussian shape
of the wave is still recognizable, and the wave’s propagation speed is close to the exact solution.
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Figure 4.
Figure 5.
Figure 6.
Figure 7. Lax Fredrichs Method
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C.
first-order upwind scheme
At t=4, the graph shows the initial Gaussian shape propagating to the right, but the wave’s amplitude is
significantly lower than the exact solution due to numerical damping, and the width of the wave is larger
than the exact solution due to numerical dispersion.
At t=10, the graph shows the wave’s propagation further to the right, but the Gaussian shape is no longer
recognizable, and the wave has spread out significantly due to numerical dispersion. The wave’s amplitude
is much lower than the exact solution, and the width of the wave is much larger than the exact solution.
Overall, the graph of the linear first-order wave equation using the first-order upwind scheme method
shows significant numerical dispersion and damping, resulting in a distorted wave that is very different
from the exact solution. This method is not suitable for solving this type of wave equation with the given
initial condition and parameters. A higher-order numerical scheme or a different numerical method may be
necessary to obtain an accurate solution.
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Figure 8.
Figure 9.
Figure 10.
Figure 11. First order upwind scheme
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D.
Second-order upwind scheme
The graph of the linear first-order wave equation using the second-order upwind scheme method shows a stable numerical solution with reduced numerical dispersion compared to the first-order upwind scheme method.
At t=4, the graph shows the initial Gaussian shape propagating to the right, maintaining its Gaussian form
with reduced numerical dispersion. The wave’s amplitude is slightly lower than the exact solution due to
numerical damping, but the width of the wave is close to the exact solution.
At t=10, the graph shows the wave’s propagation further to the right with reduced numerical dispersion
compared to the first-order upwind scheme method. The Gaussian shape is still recognizable, but its width
might be slightly larger than the exact solution due to numerical damping. The wave’s amplitude is lower
than the exact solution, but less significantly compared to the first-order upwind scheme method.
Overall, the graph of the linear first-order wave equation using the second-order upwind scheme method
shows a more accurate numerical solution with reduced numerical dispersion and numerical damping compared to the first-order upwind scheme method. The resulting wave has a slightly lower amplitude and a
larger width due to numerical damping, but the Gaussian shape of the wave is still recognizable, and the
wave’s propagation speed is close to the exact solution.
Figure 12. Lax-Wendroff method
E.
Leapfrog scheme
The leapfrog scheme is a popular method for solving wave equations, and it is known for its stability and
accuracy. In the graph of the linear first-order wave equation using the leapfrog scheme, we can observe a
stable numerical solution with reduced numerical dispersion compared to the first and second-order upwind
schemes.
At t=4, the initial Gaussian shape propagates to the right, maintaining its Gaussian form with reduced
numerical dispersion. The wave’s amplitude is slightly lower than the exact solution due to numerical damping, but the width of the wave is very close to the exact solution.
At t=10, the graph shows the wave’s propagation further to the right with reduced numerical dispersion.
The Gaussian shape is still recognizable, and its width is very close to the exact solution. The wave’s amplitude is still lower than the exact solution, but less significantly compared to the first and second-order
upwind schemes.
Overall, the graph of the linear first-order wave equation using the leapfrog scheme shows a stable
numerical solution with reduced numerical dispersion compared to the upwind schemes. The resulting
wave has a slightly lower amplitude due to numerical damping, but the Gaussian shape of the wave is still
recognizable, and the wave’s propagation speed is very close to the exact solution.
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Figure 13. Leapfrog scheme
F.
single-step Lax-Wendroff method
The single-step Lax-Wendroff method is a numerical scheme for solving partial differential equations like the
linear first-order wave equation. When applied to the given initial condition and parameters, the resulting
graph at t=4 and t=10 shows a stable numerical solution with reduced numerical dispersion compared to
other methods like the first-order upwind and leapfrog schemes.
At t=4, the graph shows the initial Gaussian shape propagating to the right with reduced numerical
dispersion. The wave’s amplitude is slightly lower than the exact solution due to numerical damping, but
the width of the wave is close to the exact solution.
At t=10, the graph shows the wave’s propagation further to the right with reduced numerical dispersion.
The Gaussian shape is still recognizable, and its width is close to the exact solution. The wave’s amplitude is
still lower than the exact solution due to numerical damping, but less significantly compared to the first-order
upwind and leapfrog schemes.
Overall, the single-step Lax-Wendroff method provides a stable and accurate numerical solution for the
linear first-order wave equation with reduced numerical dispersion compared to other methods. The resulting
wave has a slightly lower amplitude and might experience numerical damping, but the Gaussian shape of
the wave is still recognizable, and the wave’s propagation speed is close to the exact solution.
Figure 14. Lax-Wendroff method
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G.
QUICK scheme
The QUICK method is a high-resolution scheme that improves upon the accuracy of the upwind scheme by
using a quadratic polynomial interpolation to approximate the solution at each grid point. The resulting
graph of the linear first-order wave equation using the QUICK scheme with the given parameters and initial
condition at t=4 and t=10 shows a very smooth and accurate representation of the Gaussian wave profile
propagating to the right with very little numerical dispersion. The amplitude of the wave is very close to
the exact solution, and the width of the wave is well-preserved, without any visible numerical damping.
At t=10, the wave has propagated further to the right, and the profile remains very smooth and accurate
with only a slight broadening of the wave’s width due to numerical diffusion. Overall, the QUICK scheme
provides a very accurate and stable numerical solution with minimal numerical dispersion and diffusion for
the linear first-order wave equation.
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Figure 15.
Figure 16.
Figure 17.
Figure 18. QUICK scheme
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H.
Warming-Kutler-Lomax method
The Warming-Kutler-Lomax (WKL) method is not well-defined for a linear first-order wave equation like
this one. The WKL method is typically used for solving nonlinear hyperbolic partial differential equations,
which have a fundamentally different behavior than linear equations.
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Figure 19.
Figure 20.
Figure 21.
Figure 22. WKL scheme
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I.
Multi-step Lax–Wendroff method
The multi-step Lax-Wendroff method is a numerical scheme that combines the first- and second-order derivatives in time and space to approximate the solution to the linear first-order wave equation.
From the graphs obtained at t=4 and t=10, it can be observed that the wave propagates smoothly with
minimal numerical dispersion. The Gaussian shape of the initial condition is well-preserved, and the wave’s
amplitude and width are very close to the exact solution.
Compared to the first-order upwind and leapfrog schemes, the multi-step Lax-Wendroff method provides
a much more accurate and stable solution. The numerical damping is significantly reduced, and the wave’s
propagation speed is very close to the exact solution.
Overall, the multi-step Lax-Wendroff method is an effective numerical scheme for solving the linear firstorder wave equation, providing a highly accurate and stable solution with minimal numerical dispersion.
Figure 23. Lax-Wendroff method
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J.
Multi-step Richtmyer method
the stability of the numerical solution for the given problem using the multi-step Richtmyer scheme with
a Courant number of c = 0.5 and a grid spacing of x = 0.01 may or may not be stable, depending on the
behavior of the solution.
Figure 24.
Figure 25.
Figure 26. Multi-step Richtmyer method
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K.
Multi-step Beam–Warming method
As with the other numerical methods, the multi-step Beam-Warming method produces a solution to the lin2
ear first-order linear wave equation du/dt+ adu/dx= 0 for the initial condition u(x, 0) = F (x) = e−4(x−5) ,
at two different times, t=4 and t=10, with parameters c=0.5, a=2, and δx = 0.01.
Looking at the graphs, we can see that the solution exhibits some oscillations and dispersive behavior,
especially at t=10, where the oscillations are more pronounced. These oscillations are due to the numerical
instability of the method, which can occur when the time step is too large or when the spatial grid is not
fine enough.
Overall, while the multi-step Beam-Warming method can be useful in certain situations, it may not be
the best choice for simulating the linear first-order linear wave equation, especially if a high level of accuracy
and stability is required.
Figure 27. Multi-step Beam–Warming method
Figure 28. Multi-step Beam–Warming method
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L.
MacCormack method
The MacCormack method is a two-step numerical method that uses a predictor and a corrector step. The
predictor step is a first-order explicit scheme that predicts the solution at the next time step, and the corrector step is a second-order implicit scheme that corrects the prediction.
When applied to the linear first-order linear wave equation with the given initial condition, the MacCormack method produces stable and accurate solutions at t=4 and t=10, as seen in the graphs. The solution
profiles remain smooth and exhibit no signs of numerical instability such as oscillations or overshoots. The
overall accuracy of the method is comparable to that of other multi-step methods, such as the Lax-Wendroff
method and the Richtmyer method. However, the MacCormack method has the advantage of being more
computationally efficient than some of the other multi-step methods, since it uses a first-order explicit scheme
in the predictor step.
Overall, the MacCormack method is a reliable and efficient numerical method for solving the linear
first-order linear wave equation, particularly when accuracy and stability are both important considerations
Figure 29.
Figure 30.
Figure 31.
MacCormack method
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M.
Second-order ENO scheme
The multi-step second-order ENO scheme method is a high-resolution scheme used for solving hyperbolic
partial differential equations. It uses a combination of higher-order numerical schemes and a non-linear
limiting procedure to minimize numerical dissipation and dispersion. This method is very accurate and can
handle shocks and discontinuities in the solution.
For the linear first-order wave equation with the given initial condition, the ENO scheme method produces
a smooth and accurate solution with minimal numerical dissipation and dispersion. The accuracy of the solution depends on the choice of the time and space steps, as well as the value of the numerical parameters used.
At time t=4 and t=10, the solution obtained using the multi-step second-order ENO scheme method for
the given problem is highly accurate and well-behaved, capturing the evolution of the initial condition as a
traveling wave moving to the right with speed a=2.
Figure 32. Second-order ENO scheme
Figure 33. Second-order ENO scheme
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II.
A.
Answer 2
Lax–Friedrichs method
The Courant number is fixed at c = 0.5, which satisfies the CFL condition and ensures the stability of the
numerical method.
Therefore, the graph of the numerical and exact solutions for the given problem using the Lax-Friedrichs
method with a Courant number of c = 0.5 is stable. The stability of the numerical solution can be confirmed
by examining the graph of the solution and observing that it does not exhibit any unphysical oscillations or
instabilities.
Figure 34.
Figure 35.
Figure 36. Lax Fredrichs Method
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B.
first-order upwind scheme
At t=4, the graph shows the initial Gaussian shape propagating to the right, but the wave’s amplitude is
significantly lower than the exact solution due to numerical damping, and the width of the wave is larger
than the exact solution due to numerical dispersion.
At t=10, the graph shows the wave’s propagation further to the right, but the Gaussian shape is no longer
recognizable, and the wave has spread out significantly due to numerical dispersion. The wave’s amplitude
is much lower than the exact solution, and the width of the wave is much larger than the exact solution.
Overall, the graph of the linear first-order wave equation using the first-order upwind scheme method
shows significant numerical dispersion and damping, resulting in a distorted wave that is very different
from the exact solution. This method is not suitable for solving this type of wave equation with the given
initial condition and parameters. A higher-order numerical scheme or a different numerical method may be
necessary to obtain an accurate solution.
Figure 37.
Figure 38.
Figure 39. First order upwind scheme
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C.
Second-order upwind scheme
Overall, the graph of the linear first-order wave equation using the second-order upwind scheme method shows
a more accurate numerical solution with reduced numerical dispersion and numerical damping compared to
the first-order upwind scheme method. The resulting wave has a slightly lower amplitude and a larger
width due to numerical damping, but the Gaussian shape of the wave is still recognizable, and the wave’s
propagation speed is close to the exact solution.
Figure 40. Lax-Wendroff method
D.
Leapfrog scheme
Overall, the graph of the linear first-order wave equation using the leapfrog scheme shows a stable numerical
solution with reduced numerical dispersion compared to the upwind schemes. The resulting wave has a
slightly lower amplitude due to numerical damping, but the Gaussian shape of the wave is still recognizable,
and the wave’s propagation speed is very close to the exact solution.
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Figure 41.
Figure 42.
Figure 43. Leapfrog scheme
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E.
single-step Lax-Wendroff method
Overall, the single-step Lax-Wendroff method provides a stable and accurate numerical solution for the linear
first-order wave equation with reduced numerical dispersion compared to other methods. The resulting wave
has a slightly lower amplitude and might experience numerical damping, but the Gaussian shape of the wave
is still recognizable, and the wave’s propagation speed is close to the exact solution.
Figure 44. single-step Lax-Wendroff method
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F.
QUICK scheme
The QUICK method is a high-resolution scheme that improves upon the accuracy of the upwind scheme by
using a quadratic polynomial interpolation to approximate the solution at each grid point. The resulting
graph of the linear first-order wave equation using the QUICK scheme with the given parameters and initial
condition at t=4 and t=10 shows a very smooth and accurate representation of the Gaussian wave profile
propagating to the right with very little numerical dispersion. The amplitude of the wave is very close to
the exact solution, and the width of the wave is well-preserved, without any visible numerical damping.
Figure 45.
Figure 46.
Figure 47. QUICK scheme
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G.
Warming-Kutler-Lomax method
The Warming-Kutler-Lomax (WKL) method is not well-defined for a linear first-order wave equation like
this one. The WKL method is typically used for solving nonlinear hyperbolic partial differential equations,
which have a fundamentally different behavior than linear equations.
Figure 48.
Figure 49.
Figure 50. WKL scheme
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H.
Multi-step Lax–Wendroff method
Compared to the first-order upwind and leapfrog schemes, the multi-step Lax-Wendroff method provides a
much more accurate and stable solution. The numerical damping is significantly reduced, and the wave’s
propagation speed is very close to the exact solution.
Overall, the multi-step Lax-Wendroff method is an effective numerical scheme for solving the linear firstorder wave equation, providing a highly accurate and stable solution with minimal numerical dispersion.
Figure 51. Multi-step Lax–Wendroff method
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I.
Multi-step Richtmyer method
The stability of the numerical solution for the given problem using the multi-step Richtmyer scheme with
a Courant number of c = 0.5 and a grid spacing of x = 0.01 may or may not be stable, depending on the
behavior of the solution.
Figure 52.
Figure 53.
Figure 54. Multi-step Richtmyer method
J.
Multi-step Beam–Warming method
Looking at the graphs, we can see that the solution exhibits some oscillations and dispersive behavior, especially at t=10, where the oscillations are more pronounced. These oscillations are due to the numerical
instability of the method, which can occur when the time step is too large or when the spatial grid is not
fine enough.
Overall, while the multi-step Beam-Warming method can be useful in certain situations, it may not be
the best choice for simulating the linear first-order linear wave equation, especially if a high level of accuracy
and stability is required.
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Figure 55. Multi-step Beam–Warming method
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K.
MacCormack method
When applied to the linear first-order linear wave equation with the given initial condition, the MacCormack
method produces stable and accurate solutions at t=4 and t=10, as seen in the graphs. The solution profiles
remain smooth and exhibit no signs of numerical instability such as oscillations or overshoots. The overall
accuracy of the method is comparable to that of other multi-step methods, such as the Lax-Wendroff method
and the Richtmyer method. However, the MacCormack method has the advantage of being more computationally efficient than some of the other multi-step methods, since it uses a first-order explicit scheme in the
predictor step.
Overall, the MacCormack method is a reliable and efficient numerical method for solving the linear
first-order linear wave equation, particularly when accuracy and stability are both important considerations
Figure 56.
Figure 57.
Figure 58.
L.
MacCormack method
Second-order ENO scheme
For the linear first-order wave equation with the given initial condition, the ENO scheme method produces a
smooth and accurate solution with minimal numerical dissipation and dispersion. The accuracy of the solution depends on the choice of the time and space steps, as well as the value of the numerical parameters used.
At time t=4 and t=10, the solution obtained using the multi-step second-order ENO scheme method for
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the given problem is highly accurate and well-behaved, capturing the evolution of the initial condition as a
traveling wave moving to the right with speed a=2
Figure 59. Lax–Friedrichs method
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III.
A.
Answer 3
Lax–Friedrichs method
Overall, the graph of the linear first-order wave equation using the Lax-Friedrichs method shows a stable
numerical solution with reduced numerical dispersion compared to the FTCS method. The resulting wave
might have a slightly lower amplitude and a larger width due to numerical damping, but the Gaussian shape
of the wave is still recognizable, and the wave’s propagation speed is close to the exact solution.
Figure 60. Lax–Friedrichs method
B.
first-order upwind scheme
Overall, the graph of the linear first-order wave equation using the first-order upwind scheme method shows
significant numerical dispersion and damping, resulting in a distorted wave that is very different from the
exact solution. This method is not suitable for solving this type of wave equation with the given initial condition and parameters. A higher-order numerical scheme or a different numerical method may be necessary
to obtain an accurate solution.
Figure 61. first-order upwind scheme
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C.
Second-order upwind scheme
Overall, the graph of the linear first-order wave equation using the second-order upwind scheme method shows
a more accurate numerical solution with reduced numerical dispersion and numerical damping compared to
the first-order upwind scheme method. The resulting wave has a slightly lower amplitude and a larger
width due to numerical damping, but the Gaussian shape of the wave is still recognizable, and the wave’s
propagation speed is close to the exact solution.
Figure 62. Second-order upwind scheme
D.
Leapfrog scheme
Overall, the graph of the linear first-order wave equation using the leapfrog scheme shows a stable numerical
solution with reduced numerical dispersion compared to the upwind schemes. The resulting wave has a
slightly lower amplitude due to numerical damping, but the Gaussian shape of the wave is still recognizable,
and the wave’s propagation speed is very close to the exact solution.
Figure 63. Leapfrog scheme
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E.
single-step Lax-Wendroff method
Overall, the single-step Lax-Wendroff method provides a stable and accurate numerical solution for the linear
first-order wave equation with reduced numerical dispersion compared to other methods. The resulting wave
has a slightly lower amplitude and might experience numerical damping, but the Gaussian shape of the wave
is still recognizable, and the wave’s propagation speed is close to the exact solution.
Figure 64. single-step Lax-Wendroff method
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F.
QUICK scheme
The QUICK method is a high-resolution scheme that improves upon the accuracy of the upwind scheme by
using a quadratic polynomial interpolation to approximate the solution at each grid point. The resulting
graph of the linear first-order wave equation using the QUICK scheme with the given parameters and initial
condition at t=4 and t=10 shows a very smooth and accurate representation of the Gaussian wave profile
propagating to the right with very little numerical dispersion. The amplitude of the wave is very close to
the exact solution, and the width of the wave is well-preserved, without any visible numerical damping.
Figure 65.
Figure 66.
Figure 67. QUICK scheme
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G.
Warming-Kutler-Lomax method
The Warming-Kutler-Lomax (WKL) method is not well-defined for a linear first-order wave equation like
this one. The WKL method is typically used for solving nonlinear hyperbolic partial differential equations,
which have a fundamentally different behavior than linear equations.
Figure 68.
Figure 69.
Figure 70. WKL scheme
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H.
Multi-step Lax–Wendroff method
Compared to the first-order upwind and leapfrog schemes, the multi-step Lax-Wendroff method provides a
much more accurate and stable solution. The numerical damping is significantly reduced, and the wave’s
propagation speed is very close to the exact solution.
Overall, the multi-step Lax-Wendroff method is an effective numerical scheme for solving the linear firstorder wave equation, providing a highly accurate and stable solution with minimal numerical dispersion.
Figure 71. Multi-step Lax-Wendroff method
I.
Multi-step Richtmyer method
The stability of the numerical solution for the given problem using the multi-step Richtmyer scheme with
a Courant number of c = 0.5 and a grid spacing of x = 0.01 may or may not be stable, depending on the
behavior of the solution.
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Figure 72.
Figure 73.
Figure 74. Multi-step Richtmyer method
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J.
Multi-step Beam–Warming method
Looking at the graphs, we can see that the solution exhibits some oscillations and dispersive behavior, especially at t=10, where the oscillations are more pronounced. These oscillations are due to the numerical
instability of the method, which can occur when the time step is too large or when the spatial grid is not
fine enough.
Overall, while the multi-step Beam-Warming method can be useful in certain situations, it may not be
the best choice for simulating the linear first-order linear wave equation, especially if a high level of accuracy
and stability is required.
Figure 75. Multi-step Beam–Warming method
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K.
MacCormack method
When applied to the linear first-order linear wave equation with the given initial condition, the MacCormack
method produces stable and accurate solutions at t=4 and t=10, as seen in the graphs. The solution profiles
remain smooth and exhibit no signs of numerical instability such as oscillations or overshoots. The overall
accuracy of the method is comparable to that of other multi-step methods, such as the Lax-Wendroff method
and the Richtmyer method. However, the MacCormack method has the advantage of being more computationally efficient than some of the other multi-step methods, since it uses a first-order explicit scheme in the
predictor step.
Overall, the MacCormack method is a reliable and efficient numerical method for solving the linear
first-order linear wave equation, particularly when accuracy and stability are both important considerations
Figure 76.
Figure 77.
Figure 78.
MacCormack method
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L.
Second-order ENO scheme
For the linear first-order wave equation with the given initial condition, the ENO scheme method produces a
smooth and accurate solution with minimal numerical dissipation and dispersion. The accuracy of the solution depends on the choice of the time and space steps, as well as the value of the numerical parameters used.
At time t=4 and t=10, the solution obtained using the multi-step second-order ENO scheme method for
the given problem is highly accurate and well-behaved, capturing the evolution of the initial condition as a
traveling wave moving to the right with speed a=2
Figure 79. Lax-Wendroff method
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IV.
A.
Answer 4
Lax–Friedrichs method
Overall, the graph of the linear first-order wave equation using the Lax-Friedrichs method shows a stable
numerical solution with reduced numerical dispersion compared to the FTCS method. The resulting wave
might have a slightly lower amplitude and a larger width due to numerical damping, but the Gaussian shape
of the wave is still recognizable, and the wave’s propagation speed is close to the exact solution.
Figure 80. Lax-Friedrichs method
B.
first-order upwind scheme
Overall, the graph of the linear first-order wave equation using the first-order upwind scheme method shows
significant numerical dispersion and damping, resulting in a distorted wave that is very different from the
exact solution. This method is not suitable for solving this type of wave equation with the given initial condition and parameters. A higher-order numerical scheme or a different numerical method may be necessary
to obtain an accurate solution.
Figure 81. first-order upwind scheme
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C.
Second-order upwind scheme
Overall, the graph of the linear first-order wave equation using the second-order upwind scheme method shows
a more accurate numerical solution with reduced numerical dispersion and numerical damping compared to
the first-order upwind scheme method. The resulting wave has a slightly lower amplitude and a larger
width due to numerical damping, but the Gaussian shape of the wave is still recognizable, and the wave’s
propagation speed is close to the exact solution.
Figure 82. Lax-Wendroff method
D.
Leapfrog scheme
Overall, the graph of the linear first-order wave equation using the leapfrog scheme shows a stable numerical
solution with reduced numerical dispersion compared to the upwind schemes. The resulting wave has a
slightly lower amplitude due to numerical damping, but the Gaussian shape of the wave is still recognizable,
and the wave’s propagation speed is very close to the exact solution.
Figure 83. Leapfrog scheme
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E.
single-step Lax-Wendroff method
Overall, the single-step Lax-Wendroff method provides a stable and accurate numerical solution for the linear
first-order wave equation with reduced numerical dispersion compared to other methods. The resulting wave
has a slightly lower amplitude and might experience numerical damping, but the Gaussian shape of the wave
is still recognizable, and the wave’s propagation speed is close to the exact solution.
Figure 84. single-step Lax-Wendroff method
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F.
QUICK scheme
The QUICK method is a high-resolution scheme that improves upon the accuracy of the upwind scheme by
using a quadratic polynomial interpolation to approximate the solution at each grid point. The resulting
graph of the linear first-order wave equation using the QUICK scheme with the given parameters and initial
condition at t=4 and t=10 shows a very smooth and accurate representation of the Gaussian wave profile
propagating to the right with very little numerical dispersion. The amplitude of the wave is very close to
the exact solution, and the width of the wave is well-preserved, without any visible numerical damping.
Figure 85.
Figure 86.
Figure 87. QUICK scheme
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G.
Warming-Kutler-Lomax method
The Warming-Kutler-Lomax (WKL) method is not well-defined for a linear first-order wave equation like
this one. The WKL method is typically used for solving nonlinear hyperbolic partial differential equations,
which have a fundamentally different behavior than linear equations.
Figure 88.
Figure 89.
Figure 90. WKL scheme
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H.
Multi-step Lax–Wendroff method
Compared to the first-order upwind and leapfrog schemes, the multi-step Lax-Wendroff method provides a
much more accurate and stable solution. The numerical damping is significantly reduced, and the wave’s
propagation speed is very close to the exact solution.
Overall, the multi-step Lax-Wendroff method is an effective numerical scheme for solving the linear firstorder wave equation, providing a highly accurate and stable solution with minimal numerical dispersion.
Figure 91. Multi-step Lax–Wendroff method
I.
Multi-step Richtmyer method
The stability of the numerical solution for the given problem using the multi-step Richtmyer scheme with
a Courant number of c = 0.5 and a grid spacing of x = 0.01 may or may not be stable, depending on the
behavior of the solution.
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Figure 92.
Figure 93.
Figure 94. Multi-step Richtmyer method
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J.
Multi-step Beam–Warming method
Looking at the graphs, we can see that the solution exhibits some oscillations and dispersive behavior, especially at t=10, where the oscillations are more pronounced. These oscillations are due to the numerical
instability of the method, which can occur when the time step is too large or when the spatial grid is not
fine enough.
Overall, while the multi-step Beam-Warming method can be useful in certain situations, it may not be
the best choice for simulating the linear first-order linear wave equation, especially if a high level of accuracy
and stability is required.
Figure 95. Multi-step Beam–Warming method
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K.
MacCormack method
When applied to the linear first-order linear wave equation with the given initial condition, the MacCormack
method produces stable and accurate solutions at t=4 and t=10, as seen in the graphs. The solution profiles
remain smooth and exhibit no signs of numerical instability such as oscillations or overshoots. The overall
accuracy of the method is comparable to that of other multi-step methods, such as the Lax-Wendroff method
and the Richtmyer method. However, the MacCormack method has the advantage of being more computationally efficient than some of the other multi-step methods, since it uses a first-order explicit scheme in the
predictor step.
Overall, the MacCormack method is a reliable and efficient numerical method for solving the linear
first-order linear wave equation, particularly when accuracy and stability are both important considerations
Figure 96.
Figure 97.
Figure 98.
MacCormack method
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L.
Second-order ENO scheme
For the linear first-order wave equation with the given initial condition, the ENO scheme method produces a
smooth and accurate solution with minimal numerical dissipation and dispersion. The accuracy of the solution depends on the choice of the time and space steps, as well as the value of the numerical parameters used.
At time t=4 and t=10, the solution obtained using the multi-step second-order ENO scheme method for
the given problem is highly accurate and well-behaved, capturing the evolution of the initial condition as a
traveling wave moving to the right with speed a=2
Figure 99. Multi-step Lax–Wendroff method
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