burgers_validation.md

Burgers' Equation: Validation & Convergence Study

This document details the validation and convergence testing for the 1D non-linear Burgers' equation modeled in Tempest: $\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2}$

1. Validation Study & Boundary Interaction

The Periodic Boundary Trap

Our initial validation sweeps for the Traveling Shock initial condition highlighted a critical physical discrepancy when using Periodic boundary conditions on a finite domain.

• The Setup: A compression shock interpolates $u_L=2.0$ down to $u_R=1.0$ at the center of the domain.
• The Issue: Periodic boundaries force the right edge ($1.0$) to wrap around to the left edge ($2.0$). In Burgers' non-linear mechanics, a $1.0 \to 2.0$ jump acts as an expansion wave.
• The Result: The periodic expansion jump forms a rarefaction fan (a linear ramp $u=x/t$). Over time, this fan eventually interacts with and alters the isolated compression shock, so the numerical solution no longer represents the same physical problem as the analytical traveling-wave solution.

The Periodic Failure Timeline (Rarefaction Fan)

The Dirichlet Fix

To stabilize the traveling shock over long durations, we implemented Dirichlet boundaries (boundaries.Dirichlet(2.0, 1.0)). By strictly holding the boundaries at the values demanded by the infinite-domain analytical solution, we prevent the formation of the rarefaction fan.

The Dirichlet Success Timeline (Stable Shock)

Error Evolution Comparison

The numerical solution remains in close agreement with the analytical traveling-shock solution throughout the validation interval.

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2. Convergence Study

We conducted a grid convergence study across multiple direct and iterative solvers to assess their order of accuracy when resolving the discontinuous shock front.

Grid Parameters:

• $N = [250, 500, 1000, 2000]$
• $CFL (\Delta t / \Delta x) = 0.02$

Evaluated Solvers:

1. Lax-Friedrichs
2. RK4 + Upwind
3. Lax-Wendroff

Results Analysis

Solver	Observed Order ($O(\Delta x)$) in $L_2$	Theoretical Max for Discontinuities
Lax-Friedrichs	$0.26$	$0.5$
RK4 + Upwind	$0.40$	$0.5$
Lax-Wendroff	$0.68$	$0.5$

Convergence Plots (Shock)

Why are the convergence orders fractional? The theoretical convergence rate of numerical methods across a discontinuous shock profile in the $L_2$ norm drops significantly. Even in a theoretical 2nd-order scheme, the shock dominates the error, reducing the measured convergence rate well below the smooth-solution truncation order. Consequently, fractional convergence orders are expected for shock-dominated problems.

Conclusions (Shock)

1. The PDE Converges: All three solvers demonstrate positive convergence slopes (the error strictly decreases as the grid is refined).
2. Lax-Wendroff is Superior: Due to its 2nd-order predictor-corrector nature, Lax-Wendroff achieves the highest convergence rate ($O(\Delta x^{0.68})$), visibly resolving the shock front with less artificial diffusion than Lax-Friedrichs.
3. Lax-Friedrichs is Diffusive: Lax-Friedrichs exhibits heavy numerical diffusion, smearing the shock front and resulting in the lowest convergence rate ($O(\Delta x^{0.26})$).
4. Upwind + RK4: The RK4 integrator minimizes temporal truncation error, allowing the observed convergence to be dominated by the first-order spatial discretization rather than by time integration.
5. Because the solution contains a discontinuity, convergence is assessed primarily by monotonic error reduction under grid refinement rather than by recovery of the formal truncation order.

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3. Convergence Study (Smooth Profile)

To decouple the solvers' intrinsic accuracy from the discontinuity-induced bottleneck, we conducted the exact same convergence analysis using a Smooth Initial Condition (burgers_traveling_smooth with artificially high viscosity $\nu=2.0$). This completely flattens the shock into a gentle curve, allowing the solvers to demonstrate their theoretical maximum orders of accuracy.

Results Analysis

Solver	Observed Order ($O(\Delta x)$) in $L_2$	Theoretical Max (Smooth)
Lax-Friedrichs	$0.67$	$1.0$
RK4 + Upwind	$0.96$	$1.0$ (Bottlenecked by 1st-order Upwind)
Lax-Wendroff	$2.00$	$2.0$

Convergence Plots (Smooth)

Conclusions (Smooth)

By removing the shock discontinuity, the observed convergence rates closely match the theoretical truncation orders of the numerical schemes.

1. Lax-Wendroff Recovers 2nd-Order: As a true 2nd-order scheme in space and time, Lax-Wendroff perfectly attains $O(\Delta x^{2.00})$ convergence when the profile is smooth.
2. Upwind Becomes the Bottleneck: Although RK4 is 4th-order in time, the upwind spatial operator is only 1st-order. As expected, this bottlenecks the entire scheme to exactly $O(\Delta x^{0.96}) \approx 1st$-order.
3. Lax-Friedrichs Remains Diffusive: While it improves over the shock scenario, the heavy geometric diffusion inherent to Lax-Friedrichs caps it at a smeared fractional rate ($O(\Delta x^{0.67})$).