Function expressions

Every function key — in Dirichlet and Neumann conditions, body forces, and initial conditions — and the displacement key of a traveling-wave condition take a string expression that Carina compiles into a scalar function.

function: "1.0e-3 * sin(pi * x) * t"

Variable namespace

Exactly four variables are in scope, in this order:

Time is deliberately the last variable, which is what makes the symbolic time and space derivatives used elsewhere (notably the traveling-wave initial condition) well defined.

Expressions support the usual arithmetic — +, -, *, /, ^ — along with sin, cos, tan, exp, log, sqrt, abs, atan, and the constant pi.

A constant is a perfectly good expression:

function: "0.0"

Quote your expressions. YAML will otherwise interpret some of them — a bare 1.0e-3 is fine, but anything containing : or starting with punctuation is not.

Named bindings

An expression may be prefixed with name=value; bindings, separated by semicolons. The last semicolon-separated segment is the expression; every earlier segment must be a binding.

function: "a=1.0e-3; tc=2.5e-4; a*exp(-(t-tc)^2/(2*tc^2))"

Bindings are textually inlined before parsing, so the example above becomes

(1.0e-3)*exp(-(t-(2.5e-4))^2/(2*(2.5e-4)^2))

Two properties follow from this being a substitution rather than an assignment:

• Later bindings may use earlier ones. tc=2.5e-4; tau=tc/2; ... is legal; tau expands to ((2.5e-4)/2).
• Substitution respects word boundaries. A binding named tc will not match the leading t of tc*t, and a binding named a will not corrupt atan.

A segment before the final one that lacks an = is an error:

Expected `name=value` in expression binding fragment: "..."

Bindings are a readability device with no runtime cost — the substituted expression is compiled once and evaluated as a plain scalar function, with no per-call parsing.

Examples

A ramp:

function: "1.0e-3 * t"

A spatially varying hold:

function: "0.005 * (1.0 - z)"

A Gaussian pulse in time, using bindings for legibility:

function: "a=1.0e-3; tc=2.5e-4; w=5.0e-5; a*exp(-(t-tc)^2/(2*w^2))"

A twist about the z axis, applied as an initial velocity in x:

function: "omega=50.0; -omega*y"