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solve_burgers_1d

Solve the 1D Burgers equation with explicit finite differences, using an initial expression and Dirichlet boundary values; returns downsampled results and saves the full grid.

Instructions

Solve u_t + uu_x = nuu_xx with explicit finite differences.

``u_x`` uses a centred difference and ``u_xx`` a second-order centred
difference. ``initial`` is an expression in x; the ends use the constant
Dirichlet values ``left`` and ``right``. The explicit scheme requires
dt <= dx^2/(2*nu) (diffusion) and dt <= dx/max|u| (CFL); a violation is
reported as an error with a hint to increase nt. The full (t, x) grid is
saved to .npz and returned downsampled to at most 20 space and 20 time
points.

Input Schema

TableJSON Schema
NameRequiredDescriptionDefault
ntNo
nuYes
nxNo
leftNo
rightNo
t_endNo
lengthNo
initialNosin(pi*x)

Schema Changelog

Changes observed during successful MCP inspections.

  1. First observedv0.1.0

TDQS

A4.2/5.0
Behavior5/5

Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

With no annotations, the description carries full behavioral burden and does well: it discloses the explicit scheme, the two stability limits (dt <= dx^2/(2*nu) and CFL dt <= dx/max|u|), the exact error behavior (violation reported as an error with a hint to increase nt), and the side effect of writing the full grid to .npz while returning a downsampled grid. This is unusually complete for a no-annotation tool.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness5/5

Is the description appropriately sized, front-loaded, and free of redundancy?

Front-loaded with the governing equation, then discretization, boundary conditions, stability limits, and return behavior in tight sequence. Every sentence carries distinct information; there is no padding.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness4/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

With no output schema, the description correctly explains the return value (grid downsampled to at most 20x20) and the .npz side effect. The remaining gap is that the numerical/domain parameters (nx, nt, t_end, length) are left undocumented for an 8-parameter tool.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters3/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema description coverage is 0% across 8 parameters, so the description must compensate. It does clarify 'initial' (an expression in x), 'left'/'right' (constant Dirichlet values), and implicitly links nu, dx, and dt through the stability formulas, but nt, nx, t_end, and length are never defined. It covers roughly half the gap.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose5/5

Does the description clearly state what the tool does and how it differs from similar tools?

States a specific verb (solve) and resource (Burgers' equation in 1D) and gives the exact PDE u_t + u*u_x = nu*u_xx. This uniquely distinguishes it from siblings like solve_heat_1d, solve_wave_1d, and solve_pde_1d without opening any schema.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines3/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

Usage is implied by the specific PDE stated, but the description never says when to use this vs the generic solve_pde_1d or the other named solvers. The stability constraints hint at applicability but are not framed as when-to-use guidance, and no exclusions or alternatives are named.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.

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