Advection Equation Solver Calculator

Computed Result

Shown above the form after submission.

Exact value:
0.367879

Phase shift:
1.200000

Courant number:
0.200000

Best method:
Lax-Wendroff

Upwind: 0.417041 | error 0.049162

Lax-Friedrichs: 0.401520 | error 0.033641

Lax-Wendroff: 0.378218 | error 0.010338

Status: Stable Courant range. Numerical steps used: 60.

Initial profile

Evaluation position x

Evaluation time t

Advection velocity c

Amplitude

Baseline

Profile center

Width or spread

Wavelength

Left edge

Right edge

Domain length

Grid points

Time step

Use periodic boundary wrapping

Example Data Table

Sample nodes compare the initial field, exact transport, and three common finite difference estimates.

x	Initial	Exact	Upwind	Lax-Friedrichs	Lax-Wendroff
0.0000	0.000285	0.000000	0.000006	0.003964	0.000010
1.5000	0.600373	0.002745	0.011283	0.094787	0.001335
3.0000	0.129923	0.921610	0.803105	0.530335	0.935988
4.5000	0.000003	0.031778	0.072595	0.197247	0.036481
6.0000	0.000000	0.000000	0.000013	0.004822	0.000001

Formula Used

Linear advection equation: u_t + c u_x = 0

Exact transported solution: u(x,t) = f(x - ct)

Grid spacing: Δx = L / (N - 1)

Courant number: λ = c Δt / Δx

Upwind scheme: uⁿ⁺¹_i = uⁿ_i - λ(uⁿ_i - uⁿ_i-1) for positive c

Lax-Friedrichs: uⁿ⁺¹_i = 0.5(uⁿ_i+1 + uⁿ_i-1) - 0.5λ(uⁿ_i+1 - uⁿ_i-1)

Lax-Wendroff: second-order transport update using centered slope and curvature correction.

The exact solution shifts the initial profile without changing shape. Numerical schemes approximate that transport and may introduce diffusion or oscillation.

How to Use This Calculator

Choose an initial profile such as Gaussian, sine, square, or triangle.
Enter the position, time, advection velocity, and shape parameters.
Set domain length, grid points, and time step.
Enable periodic wrapping if your domain behaves cyclically.
Press Solve Equation to show the result above the form.
Review exact and numerical values, then inspect errors and stability.
Download CSV for raw values or PDF for a clean report.

Frequently Asked Questions

1. What does this solver compute?

It evaluates the linear advection equation at a chosen position and time. It also compares the exact transported profile against upwind, Lax-Friedrichs, and Lax-Wendroff numerical approximations.

2. Why is the exact solution just a shift?

For constant velocity advection, the initial shape moves rigidly along characteristics. The profile value at x and t equals the initial value located at x minus ct.

3. What is the Courant number?

The Courant number is cΔt/Δx. It measures how far information moves during one time step relative to grid spacing. Many advection schemes are reliable only when its magnitude stays at or below one.

4. Which method is usually most accurate?

Lax-Wendroff often gives sharper results for smooth profiles, while upwind is more diffusive but robust. The best choice depends on profile shape, grid resolution, and stability limits.

5. Why do square pulses look smeared?

Numerical diffusion smooths steep fronts, especially with first-order schemes like upwind and Lax-Friedrichs. Finer grids and smaller time steps can reduce smearing but will not remove it completely.

6. When should I use periodic boundaries?

Use periodic boundaries when your domain wraps around, such as a repeating ring or cyclic interval. A transported feature exiting one side re-enters on the opposite side.

7. What do CSV and PDF exports include?

The CSV export includes key metrics and sample solution rows. The PDF export creates a print-ready report of the page, including results, formulas, instructions, and the example table.