Solving 1D Linear Advection Equation and Burgers’ equation using Finite Volume Method

Submitted By: Ajit Nepal

Reg.No:22UMPY01

Under The Supervision of

Dr. Rupak Mukherjee

Department of Physics

School of Physical Sciences

Sikkim University

Analytical Solution of Advection Equation

1D Linear Advection Equation (Wave Equation)

• Definition:

Described by: Here, u(x,t) is a scalar (wave)

Velocity c during time t and f=f(u).

For f=cu

• Wave Propagation:
• Direction:
• If : Wave propagates in the positive x-axis direction.
• If : Wave propagates in the negative x-axis direction.
• Speed:
• The magnitude of c determines the speed of wave propagation.

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Exact Solution

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Given the initial condition

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c=1

After time t:

Numerical Methods for the Linear Advection Equation

Two popular methods for performing discretization:

• Finite Differences
• Finite Volume

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For some problems, the resulting discretizations look identical, but they are distinct approaches.

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We begin using finite-difference as it will allow us to quickly learn some important idea

Finite Difference Method

• A finite-difference method stores the solution at specific points in space and time.

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• The spatial domain is discretized, or broken into a finite number of steps or grids.
• Associated with each grid point is a function value
• We replace the derivatives in out PDEs with differences between neighboring points 
• FDM is basically of three types : forward , backward and central difference.

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FTCS Method (Forward in Time, Centered in Space)

Plots Of FD and understanding the results

Highly unstable Amplitude

Solution is Dissipative in Nature

Under Multiple Revolution around the Domain

-Amplitude is Decaying

Finite Volume Method

• Key Points of the Finite Volume Method (FVM)
• Definition and Usage:
• The Finite Volume Method (FVM) is used to represent and solve partial differential equations (PDEs) as algebraic equations.
• Widely applied in computational fluid dynamics (CFD) packages.
• Fundamental Principles:
• Volume to Surface Integrals:
• Converts volume integrals with divergence terms to surface integrals using the divergence theorem.
• Flux Evaluation:
• Evaluates terms as fluxes at the surfaces of each finite volume.
• Ensures conservation by making the flux entering a volume equal to the flux leaving the adjacent volume.
• Unstructured Mesh Compatibility:
• Easily formulated for unstructured meshes, aiding in the modeling of complex geometries.

Working of Finite Volume Method

Governing Equation

Take volume average of the governing equation at t=t2 and apply Divergence theorem to get Semi-Discrete Numerical Scheme

Control Volume

Constant Rule:

Trapezoidal Rule:

Midpoint Rule:

Simpson's Rule:

• Fundamental Building Blocks in Spatial Discretization
• Calculate cell average within the Control Volume
• We have the Equations that represent the fluxes entering and leaving the control volume.

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• equations that represent the fluxes entering and leaving the control volume.

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Numerical Integration: For calculating Cell Average

Upwind Method�

With the approximation on the right, We modify the iteration equation as

Positive Flow Direction (c>0)

Now that we have calculated cell average next step is to find Determine the values of u at the faces of the control volume.

Negative Flow Direction(c<0)

Plot showing advection of sine wave using upwind Method

-Amplitude starts decaying as the simulation time increases

Plots and Discussion

• Key Takeaway from the plots:
• solution starts to dissipate as the simulation time is increased

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• Starts to smooth out sharp features and slopes for the original condition with sharp corners
• It is not accurate enough to be of any use for actual simulation tasks

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Plot of U vs X for Linear Upwind Scheme with Finite volume. Initial condition sin(2πx) and Square�wave respectively�

-Stable for few rounds

-Dissipative in Sharp Edges

Lax-Wendroff Method

Achieves second-order accuracy in both space and time

Algorithm:

Key Features:

• Accuracy: Second-order in both space and time, enhancing resolution of waveforms and reducing numerical diffusion.
• Can capture the structure of model quite accurately and carry it

Cons:

• Stable for selective cfl values
• Complex algorithm to implement
• Computational Cost

• k1​: Initial slope at the current point.
• k2k_2k2​: Slope at the midpoint using k1
• k3k_3k3​: Slope at the midpoint using k2.
• k4k_4k4​: Slope at the next point using k3.
• Update formula: Combines these slopes to give the new value.

RK4 Method :

• RK4 offers a good balance between computational complexity and accuracy.
• Involves more calculations per step compared to first-order methods

• Initial condition: sin(2πx)

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• Number of revolution:100
• Stable and No unwanted oscillations�

• Initial condition: Gaussian Function, with amplitude 1 unit
• Number of Revolution: 1000
• Stable and no unwanted oscillations

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Plots and Discussions:

• Initial condition: Square Wave Form
• Very Stable
• Some unusual peaks and oscillatory�behaviors around the points where the wave has sharp discontinuities

Flux Limiters

• Low-order schemes are stable but dissipative near�discontinuities
• Higher-order schemes are more accurate but prone to�oscillations
• The challenge is to create schemes that are both accurate and oscillation free, known as high-resolution scheme
• using flux limiter functions, we can blend high-order and low-order numerical fluxes.
• Smooth Region--🡪High Order
• Discontinuous Region 🡪 Low Order

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Working of Van Leer Limiter Function

Numerical Flux with Van Leer Limiter

Ratio of Successive Gradients r:

Unconditionally Stable Plot

Applying our High-Resolution scheme to inviscid Burgers’ Equation

Analytical Solution

Numerical Solution

Governing Equation

Breaking time

Is The moment when the solution develops a shock or discontinuity

Conclusions

High Resolution Schemes

Precision: Excellently captures discontinuities and steep gradients.

• Robustness: Effective in complex, nonlinear scenarios.
• Conservative: Maintains essential physical properties like mass and energy.
• Versatile: Compatible with diverse mesh types and complex geometries.

Challenges:

Costly: Demands significant computational resources.

• Complex: Requires advanced algorithms for accurate simulations.
• Stability: Needs careful handling at high dynamics to ensure stability.

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Applications:

Aerospace: Simulating shock waves and airflow around aircraft.

• Automotive: Designing for aerodynamics and thermal efficiencies.
• Environmental: Modeling pollution spread and weather systems.
• Energy: Analyzing flows in turbines and complex energy systems.
• Biomedical: Studying blood flow dynamics and effects of interventions.

Thank You