You could try something like:
$('.open-everything').click(function() {
$.each($('.accordion-title'), function(index, value) {
if (!$(this).hasClass('open')) {
$(this).trigger('click');
}
});
});
And then have a button with the classname of open-everything:
<button class="open-everything">Open Everything</button>
Make sure to set autoclose=false on your open accordion shortcode.