Solution 1:

As discussed in the jQuery UI forums, you should not use accordions for this.

If you want something that looks and acts like an accordion, that is fine. Use their classes to style them, and implement whatever functionality you need. Then adding a button to open or close them all is pretty straightforward. Example

HTML

By using the jquery-ui classes, we keep our accordions looking just like the "real" accordions.

<div id="accordion" class="ui-accordion ui-widget ui-helper-reset">
    <h3 class="accordion-header ui-accordion-header ui-helper-reset ui-state-default ui-accordion-icons ui-corner-all">
        <span class="ui-accordion-header-icon ui-icon ui-icon-triangle-1-e"></span>
        Section 1
    </h3>
    <div class="ui-accordion-content ui-helper-reset ui-widget-content ui-corner-bottom">
        Content 1
    </div>
</div>​

Roll your own accordions

Mostly we just want accordion headers to toggle the state of the following sibling, which is it's content area. We have also added two custom events "show" and "hide" which we will hook into later.

var headers = $('#accordion .accordion-header');
var contentAreas = $('#accordion .ui-accordion-content ').hide();
var expandLink = $('.accordion-expand-all');

headers.click(function() {
    var panel = $(this).next();
    var isOpen = panel.is(':visible');

    // open or close as necessary
    panel[isOpen? 'slideUp': 'slideDown']()
        // trigger the correct custom event
        .trigger(isOpen? 'hide': 'show');

    // stop the link from causing a pagescroll
    return false;
});

Expand/Collapse All

We use a boolean isAllOpen flag to mark when the button has been changed, this could just as easily have been a class, or a state variable on a larger plugin framework.

expandLink.click(function(){
    var isAllOpen = $(this).data('isAllOpen');

    contentAreas[isAllOpen? 'hide': 'show']()
        .trigger(isAllOpen? 'hide': 'show');
});

Swap the button when "all open"

Thanks to our custom "show" and "hide" events, we have something to listen for when panels are changing. The only special case is "are they all open", if yes the button should be a "Collapse all", if not it should be "Expand all".

contentAreas.on({
    // whenever we open a panel, check to see if they're all open
    // if all open, swap the button to collapser
    show: function(){
        var isAllOpen = !contentAreas.is(':hidden');   
        if(isAllOpen){
            expandLink.text('Collapse All')
                .data('isAllOpen', true);
        }
    },
    // whenever we close a panel, check to see if they're all open
    // if not all open, swap the button to expander
    hide: function(){
        var isAllOpen = !contentAreas.is(':hidden');
        if(!isAllOpen){
            expandLink.text('Expand all')
            .data('isAllOpen', false);
        } 
    }
});​

Edit for comment: Maintaining "1 panel open only" unless you hit the "Expand all" button is actually much easier. Example

Solution 2:

A lot of these seem to be overcomplicated. I achieved what I wanted with just the following:

$(".ui-accordion-content").show();

JSFiddle